 So once again, good afternoon to those who have joined in. So today, we are going to start with a new concept in the bridge course, which is your introduction to calculus. Okay, graphs and all, even though there are many other graphs to be done, many other function relation graph to be plotted, we are going to take that once we reopen after your board exams happen. Okay, so don't worry about missing anything, even though your assignment worksheet says so many different functions, right? You would be seeing some square brackets, some curly bracket type of thing, log exponential, maximum. So we'll be taking that up later on. Don't worry about it. There's nothing like we're going to skip because all those are very, very integral part of your learning in mathematics. So, see if this is a class on Friday, physics because we have exam on Saturday. No, you will not have an exam, not have a class on Friday, I think sir will drop you a message a day before. I kept this class because Kormangala people, your next exam is on 3rd of April. So you have enough time. So you have two days left for that. So that's why we kept the class. So, sir, so about classes are over till boards. AB classes, I didn't get that. Oh, you wrote that in Hindi. Yes, kind of, kind of, because most of you are going to start with your board exams in the month of in the first week of May, so it would be over mostly. Okay, so today we're going to start with introduction to calculus, introduction to calculus, introduction to calculus, calculus. Now, most of you would have heard of this term calculus, maybe from your seniors, maybe from your elder brothers or sisters, right, or maybe people talking around you. Okay, and some of the students who were there with us in the accelerator program last year, Tushar sir would have definitely taken you through calculus, a bit of calculus at least. So calculus is something which has been a buzzword in mathematics. Okay, and rightly so because it is one of the major breakthroughs in the field of mathematics. Calculus personally I feel that this this branch of mathematics was actually invented. Okay, so see when mathematics was evolving. Okay, it evolved from geometry. So geometry was the first field of mathematics that most of us, in fact, the entire world studied geometry. So we talked about shapes, circles, triangles, right. So, even a small child who starts learning mathematics, he starts his mathematical journey with shapes, right, my daughter, who's just four and a half years, she's been taught about what's a circle, what's a triangle and all those stuff. So later when math evolved from geometry, we went to algebra, where we started talking about numbers, we started talking about equations and all those algebraic operations. And then we started moving towards analytical geometry, what we know as the coordinate geometry. So our first chapter of the bridge course, where we learned about graphs and its and its transformations, it was basically based on analytical geometry. Now calculus is a very, very new field of mathematics, even though it was invented in the 17th century by two famous scientists. Okay, independently of each other. So it was invented by Isaac Newton and Gottfried Wilhelm Levenies. I'll just write his last name. It is a big name. This guy is a German guy. So Newton everybody knows, right, he has been very instrumental in the field of gravitation in the field of optics. Levenies is a German guy, Gottfried Wilhelm Levenies, who actually independently invented calculus. Actually Levenies was accused of plagiarism by Newton. But later on it was proved wrong. It was basically found out that Levenies had almost in the very same time frame came up with the same set of ideas which actually Newton had developed. Now I'll just go into a brief history about calculus because it's very interesting for you to know that calculus actually comes from a Latin word, Calculi. Okay, Calculi is basically a word for pebbles, pebbles were used for counting. Okay, so calculus was developed or calculus was invented when a very rich, you can say, an astronomist by the name of Edmund Haley. He visited Newton and he said Newton was at that time a well-renowned figure. So Edmund Haley, who was a very rich guy, he asked Newton to work upon the gravitation force between two celestial bodies. So two planetary bodies, he asked him to make a study about it. Okay, so what he said was what he asked Newton to do. So let's say there are two celestial bodies. So I'm just making one with red, other with yellow. Okay, and let's say between these two celestial bodies, the distances are. Now, of course, we all know that there is a force of attraction between two celestial bodies. Correct? So Edmund Haley wanted Newton to work on how does this force change when the distance between the two celestial bodies changes. Okay, so let's say now it is r and let's say it becomes r plus delta r. Okay, delta r is basically to signify the change in the distance. So what is this, what impact does it bring on the force of attraction between the two bodies? So let's say because of the change in r to r plus delta r, the force of attraction is also going to change, isn't it? So the bodies are going to go faster and go away from each other. The force of attraction will reduce. If they come closer, the force of attraction will increase. Correct? So he wanted him to study how does the force of attraction change with respect to change in the distance. Okay, and when this change is very, very small, okay, right now I have used a symbol over here delta r and I put an arrow and zero. Okay, so when he approached Newton to make a study of this, okay, he wanted him to tell him how does infinitesimal change in the distance between their centers bring about a change in the gravitational force of attraction. Okay, so a quantity which is very small like this, there is a new word that you will get to learn today and that word is infinitesimal, infinitesimal. Infinitesimal is basically made up of two words, infinite plus small. So infinite and small, if you combine it becomes infinitesimal. So understand this. So basically Edmund Haley, I think Haley word you would have heard of before, has anybody heard of the word Haley before? Let me check your geography skills. Haley word, anybody has heard of it? Haley's Comet, exactly. Arjun Damesh, sir is this class for NPS Indranagar. No dear, it is not for Indranagar, have people from Indranagar also joined in? Anybody from Indranagar has also joined in? I don't think so, this message was sent to Indranagar. So Arjun, how did you enter the session? Okay, okay, so no, no, this is not for Indranagar. Indranagar session will happen independently but if you want to be a part of the session, you are most welcome, Arjun. Okay, so any Indranagar student who has joined in this session, you are most welcome to stay in the session. Okay, yeah, because you also missed a class because you had exams going on. So if you want to be a part of the session, you can be there. But yes, the same thing will be repeated again for your batch as well once your exams are done. Okay, anyways, let's continue. Right, so now this branch of mathematics, as it suggests from the simple example is basically deals with quantities which are changing continuously. That means there is an infinitesimal change in the quantities. How does it affect another quantity related to it? So here what was the case? Basically, here was a case where one quantity, which is your force is related to the distance between the bodies. So how does this force change? If there is a small change in our that is what calculus helps us to figure out. Okay, so many people say, sir, what is the role of calculus here? So calculus is nothing but it's a field of mathematics. It's a vertical of mathematics, which helps you to understand how does one quantity change with respect to another when there is a very, very small change, infinitesimal change, or there is a very continuous change happening in the other. So see these two bodies, because of their rotation, revolution, whatever, the distance between them will gradually change. It doesn't happen abruptly. It is a gradual change. So if it is gradually changing, how does the force between them changes? That is what calculus helps us to figure out. Okay, so in plain and simple words, if you ask me, what does calculus help us to study? Calculus is basically a branch which helps us to study continuous changes in variables. Okay, this term, later on you would be learning it is called DF by DR. This is what will be called as DF by DR, where this D stands for differential, differential. Differential means a change which is as good as zero change, almost zero change. Aryan, sir, my third pre-boot got over today. So can I send my homework by tomorrow? Yeah, Aryan, don't worry about homework. Okay. Okay, yeah. Okay. Now, for God's sake, please do not cancel off DD, okay. D stands for differential change. So Newton was very influential towards what we call as differential calculus. So this will be studied under, this would be, this part of calculus would be studied under differential calculus, differential calculus. So are there different verticals of calculus as well? Yes. What you know is just, you know, the broader version of calculus. So Newton was effectively working on something called differential calculus. So he wrote a book called method of, method of fluctuations. Okay, this fluctuations word later on was renamed to derivatives. Okay, derivative is something which comes out from the process of differentiation. Okay, so not a lot of new words coming your way, especially for people who have not heard about it. So Newton, just understand as of now that Newton was effectively working on differential calculus. Okay, differential calculus is where we deal with derivatives. What is derivative? We'll talk about it later on. Don't worry about it right now. Okay, Leibniz on the other hand was instrumental towards working on integral calculus. Okay, so he worked on integration mainly. Okay, he worked on integration part, but both of them basically worked on calculus together. And whatever they worked on, they were both, they are basically linked to each other. Okay, and these two are linked to each other. So they are linked by fundamental theorem of calculus, fundamental theorem of calculus. What is it? Don't worry about that now. Just understand that Newton was working on differential calculus, Leibniz was working on integral calculus and both these fields of calculus are linked to each other by a theorem called fundamental theorem of calculus. Okay, what is the details of these, you know, small, small concepts which I've given to you, we will talk about it in some time. Don't worry about it. Okay, so by this time, you should have understood who invented calculus, correct? It was independently done by Newton and Leibniz. And what is this calculus actually trying to study? So I'm just giving you one example of it. So here probably I give you one differential calculus example. Let me give you more examples. So let us understand why calculus becomes so important in the field of maths, physics and chemistry, which all the three are going to study this year. So let us understand what is the use of calculus. Okay, so a few examples just to understand what is the use of calculus for us. Okay, now let me give you a simple example just to illustrate what is the use of calculus. Let's say you are going from Bangalore to Mysore on your car. Okay, so let's say along the y-axis we plot the displacement of your car and on the x-axis we plot the time taken by your car. Okay, so let's say you started your journey or let me put it over here. You started your journey at t equal to 0 and here you are at Bangalore. Okay, so at t equal to 0 when you started your journey you are at Bangalore. Okay, and let's say by t equal to 5 you reached Mysore. Okay, and Mysore is let's say here. And the journey of your car looks like this. Okay, so by this time you reached your destination which is Mysore. Okay, and let's say Mysore to Bangalore this distance is about 150 kilometers. Okay, and you took 5 hours. Let's say this time is in hours and the displacement is in kilometers. Now let me ask you a simple question. What is the average speed of your car? What is the average speed of your car? Who will tell me? A bit of physics I am asking you over here. Exactly Sharguli. Exactly. I think nobody has any doubt Karthik absolutely correct. So basically what you did, you basically took total change in the displacement by total change in the time. Right, so what was the change in the displacement? 150 kilometers correct. So from Bangalore to Mysore the change in the displacement that happened was 150 kilometers. What was the change in the time? Change in time was 5 hours correct. So when you divide 150 kilometers by 5 hours you get your answer as 30 kilometers per hour. Okay, this is what we call as the average speed of the car. Okay, nobody has any doubt in that. Any doubt? Okay. Now let us say I ask you another question on the same. Look at this graph. Look at this graph using this graph. Tell me what was the speed of your car at t equal to 3 hours? What was the speed of your car at t equal to 3? Many of you would be like what? Just at that t equal to 3 what is the speed you are asking? Because see many people are baffled by this question. Many people are shocked by this question because I have not mentioned any time interval of travel. I am asking you at a pinpoint time at t equal to 3. Let's say you started your journey at 12 noon. So when the clock struck 3pm let's say you are somewhere here on your journey. Correct? So this is let's say t equal to 3. Now here if I ask you at this instant of time what was the speed of your car? How will you solve this question? What will that give you Ashna if you drop perpendicular on the x-axis and the y-axis? How does it help you? Doesn't it? Now I don't see any answer coming from any one of you because we can't say close to this. So now Sharduli is saying let's allow the car to move for 2 seconds and then we will see how much distance it covered or how much displacement it had. So you take the displacement divided by 2 seconds. But why 2 seconds? It will be giving you the average velocity between 3 to 3 hours 2 seconds. But why 2 seconds? Why not 1 second? Why not 0.1 second? Why not 0.001 second? Why are you allowing the car to move for 2 seconds? 2 seconds may be a very large time. Correct? Now you have to allow your car to move for a infinitismal time. That is the concept that you are basically trying to apply over here. The word infinitismal the moment it comes basically calculus will jump at that instant of time. So here what are you doing in order to know what is the speed of the car at t equal to 3? You need to allow the car to run for a very small time and that time is almost tending to 0. And you are trying to see what is the displacement of the car in that point of time. Are you getting my point? So in the neighborhood of t equal to 3 you have to allow the car to move for a very very small magnitude or very very small time duration which is what we call as infinitismal time duration. And you need to check what is the displacement in that time and this will give you the velocity of the car at t equal to 3. Now such concepts here is what we call as finding the differential or finding the derivative of the displacement with respect to time at t equal to 3. As you can see calculus has featured over here. This is what we call as instantaneous speed. So whenever instantaneous concepts come into picture calculus has to be used. Are you getting my point? So we cannot solve this question till we know what is the use of what is the calculus here. So calculus basically will help you to know that at this instant of time what is the velocity. I am not asking you what is the displacement at that time. I am asking you what is the velocity at that time. Are you getting my point? Later on we will learn that the velocity can be obtained by knowing the slope of the tangent at this point. So if you sketch a tangent to this curve this tangent will give you, let me write it down. The slope here will give you the speed instantaneous speed at t equal to 3. Now how we will answer this little later on. Don't worry about it. So now this is just one example. So here you would see that we are primarily using differential calculus. So can I give some example where integral calculus will be used. So let me give you another instance. So let's say I ask you this question. There is a line segment which is connecting two points A whose coordinate is X1, Y1 and B whose coordinate is X2, Y2. Tell me what is this distance AB? Siddharth has a question. Sir if the speed of the car changes in time how can we find the speed? See speed of the car is changing every instant. So I am asking you exactly when t was 3 what was the velocity or what was the speed of the car? At that instant. At a particular instant it will have one speed, right? Siddharth. Okay so I am looking for that speed. Right, most of you have given the answer to this. The answer to this is X2 minus X1 or X1 minus X2 whole square doesn't matter because you are squaring this stuff. Okay, now how do you get this formula by Pythagoras theorem, right? So basically geometry was sufficient for you to know the answer for this. So what you normally do is you consider that there was a coordinate axis. I will just make a small coordinate axis over here. So let's say this was a coordinate axis for you, your Y axis and your X axis. Okay, so here I normally make a line parallel to the X axis and I make this line parallel to the Y axis. Okay, now when you are moving along a line which is parallel to the X axis, remember Y coordinate will not change. So here you can see Y coordinate will be Y1. Okay, and if you move along a vertical line, X coordinate doesn't change. So here your coordinate will be X2. Right, abscissa will be X2. So AC distance you can all easily figure out is mod X2 minus X1. BC distance you can easily figure out is mod Y2 minus Y1. Okay, and you basically know your Pythagoras theorem that AB square is equal to AC square plus BC square. AC square plus BC square. And from here you end up getting X2 minus X1 mod square. Remember squaring will just take off that mod thing. You don't require a mod there. And this is your AB square. So if you just remove the square by taking a square root, there you get the distance. So here geometry was sufficient for you to find the answer. Correct. All you required was your Pythagoras theorem. But what if I change my question to this? So there is a curve, okay, which looks like this, you know. Okay, so this is some curve which I know Y is equal to R of X or F of X, whatever you want to call it. Correct. So tell me what is the length of this curvilinear line. Curvilinear means it is a curve kind of a line. Okay, so what is this distance? How will you find out? How will you find out? See, we all have visited Bangalore. Bangalore is like, you can say a city of flyovers, right? So there are some flyovers. I visited one of the flyovers in Indranagar. It was like this, like this. How do I, of course I can go with a string around the flyover and get the length of the string. But let's say if I know the equation of this, you know, path, can I get the length of this path? Can I get the length of the sign segment? But will it be accurate, Sharduli? That's my question. Won't there be glitches? Won't there be, you know, errors coming into your answer? Right, there would be. So how do I find exact answers to these particular situations? Because the line segments which you are using, it should be infinitesimally small, small line segments, isn't it? Yes or no? And you are accumulating those line segments to know what is the length of this particular path from A to B. Now the word infinitesimal has come again here, correct? And you are accumulating it to get the entire length of the line segment AB. Now the word accumulating is basically linked to integral calculus. Okay. So the answer to this, I'll just write it down. But don't expect me to give you an explanation for it right now because you're not aware of the basics. So just to give you an idea, the answer for this would be, let's say this is your x equal to A. And this point is x equal to B point, okay? The answer to the length of line segment AB would be integration of under root of 1 plus derivative of R with respect to x whole square. Into dx integrated from A to B. Okay. Now don't worry. This is like, you know, it went above your head. I understand that. It went tangentially across your head. But don't worry about it. We will talk about it later on in more detail. What I wanted to explain here was what was my main aim of writing such thing is what? Here you are using integral calculus. Okay. And this symbol, you are going to see a lot. This symbol, the snake. Okay. By the way, let me tell you some small story about the snake. This snake was actually, it's a symbol for integration. It's a symbol for integration. Okay. This was invented by this symbol was given to us by Lebanese. Okay. So Lebanese was the one who gave us this symbol. The symbol is actually a elongated s in long gated s where s basically stands for summation. Oh yeah. This is X my dear. But don't don't worry about it. Say it right now because even if you note down the expression, you don't know how it comes actually. Okay. So that we have to understand calculus in more depth. Anyways, so this guy. See, what are you doing? You're doing accumulation. Correct. You are accumulating the length of those small, small, small, small line segments what Sharduli was talking about. Right. So the art of accumulating these small changes is what is actually done by integral calculus. So what does differential calculus do then differential calculus studies the change in a quantity with respect to another when that another quantity is changing very, very minutely or very, very infinitesimally or very, very continuously. Are you getting a point? So just remember the example of the Newton working on how does the gravitational force between the two planetary bodies change with respect to their small changes in the displacement. So when he's studying that, he's basically studying DF by DR, which is a part of differential calculus. Whereas, whereas our friend Lebanese was working on accumulative calculus. Okay. Where you're trying to add those small, small changes to make up a big quantity. Okay. So integration, the word itself, literal meaning of the word integration means combining, adding. Right. Isn't it? Isn't it? The integration means that. So when we start combining these small, small, small, small quantities to give large to a bigger quantity, we are basically performing the operation of integration. Earth. What is earth? Earth is an integration of dust particles. Correct. So when dust particles accumulate, we basically get our. Correct. Isn't it? So all you need to understand from this example is that you will not be able to find answers to such questions without knowing calculus. Correct. Let me give you one more example. Let me give you one more example. Let's say I ask you this question. There is a curve or there is a straight line like this. Where is my white marker? Yeah. So let's say there is a straight line like this. Okay. Okay. So this line is basically y is equal to a line. Fine. Now let's say this is x equal to zero. And this point is x equal to b. Correct. Can you tell me what is the area which is trapped between the x-axis and this line? I'm sure you would not even take a second to answer this. Right. There you go. I'm getting the answers a, b. Absolutely. Right. Because all you could figure out here was it's a rectangle with length as b height as or with as a and you basically multiplied your length and width and got your answer. But what if I change this question? Slightly. Now let's say instead of having a line, straight line like this, your line was something like this. Okay. Right. But you knew what was this function. Let's say this function or this relation is known to you. Now tell me. Well, I made it very long. Okay. Now tell me what do you think is the area trapped between this curve and the x-axis between zero to b. What is this area? Now I'm sure you would be scratching your head. Are you? How do I do this? Correct. Right. Yes or no. But let me tell you the answer to this is integration of the function multiplied with dx from zero to b. This gives you the area. Okay. Now just to give you a brief idea, how does this come out? So what do we do normally? We at a distance of x. Okay. We consider a small differential change in x, which is called dx. Correct. Now this height, which you have taken at a distance of x, this height will be your f of x. So what do we do? We construct a thin triangle, thin rectangle over here. A rectangle like this. There you can see a rectangle being framed. Okay. So roughly speaking, this is a rectangular structure, which we are making over here. Let me just shade it. Okay. Now this rectangle will have an area of f of x into d of x. Correct. So to know the full area, you have to keep adding these small, small area. Correct. This kind of area, the strip area, you need to keep adding from zero all the way till b. So that is what is written by integration from zero to b. Integration means you're summing it up from zero to b. So when you keep, see if you take a clock, let's say a handkerchief. A handkerchief is basically what it is made up of, you know, heads, isn't it? So let's say this thin strip is like one of the threads of that handkerchief. So when you add all those thread area, basically it gives you the area of the full handkerchief. Right. So that is what here we are doing precisely. So this strip that you see, you can treat it as if it is a thread in your handkerchief and you are trying to add all the areas of those thread to make up area of the handkerchief. So again, you are using accumulative calculus, which is what is done by integral calculus. Okay. How to do this process? Again, we will take it up later on. Don't worry about it. Sharduli has a question. So do we have to take the middle area or we can take smaller triangles at the beginning? See, this is just, what did I say? At any distance x, you are taking this just because it is looking in the middle. It is not in the middle. At any distance x. So basically what you're doing in general, you are going at a distance x picking up a strip. You're finding the area of that strip and you're adding all the way from zero till the end where you need it. Okay. So it is not in the middle. Though it is looking like that, it is not in the middle. Got it? Okay. So let us begin the process of learning now that we have known why calculus is important. Let us start learning it. Okay. So the other processes I have explained to you, though they were tangential to you as of now, we will try to understand them to depth, whatever time permits us. Okay. Now, let me tell you calculus is something which you will immediately need in physics, not in maths. Maths, it will be taken up in your school by the, you can say, middle of the session early. Say if your session starts in the June, July, August, September, October timeframe, it will be taken up. But in physics, you're going to start with calculus. I think from the second or third class, right, right after your units and dimensions, the moment your teacher starts kinematics for you, you will have to use calculus. Okay. And let me tell you most in most of the cases, physics teachers do not go into details of calculus because they're mostly interested in their application part. Okay. So as a math teacher, it becomes my duty. It's my responsibility to make you aware of these nuances, which we're going to face sooner or later, sooner in physics, later in maths. So in chemistry, is it also used? Somebody is asking. Yes. In chemistry, also it is used when you are studying chapters like chemical kinetics. Right. And in physics, when you're studying about the order of a reaction, you will be studying calculus there as well. You will be using calculus there as well. Okay. So so many uses of calculus. The other day some, I met a student. I just wanted to share this story with you. The student said that sir, she's, the student is in class 10th right now. And in class 11th, he wants to drop maths. Okay. Probably he has got some kind of what do you call brainwashing from a senior that maths is difficult though it is nothing like that. So he said, I want to drop maths. I said, why, why you want to drop maths because almost every field needs it nowadays. No sir, I want to do economics. Right. So maths is not required there. I said, who told you in economics, maths is not required. Economics is full of how one quantity changes with respect to other. And that is differential calculus. Correct. Let's say you are a person, you are an IAS officer. Let's say, okay, you are into the decision making, you know, body of the government. You want to know how does the price of rice gets changed when there is a one passage change in the price of diesel or kerosene. You have to apply differential calculus there. Isn't it. So an economist needs calculus more than anybody else. Right. So please note, do not be under this impression that economics and other fields of maths do not need calculus. Calculus is you can say only present. It is present almost in every field of maths. Commerce students have to study calculus. Forget about science students. Science students have to study calculus. No doubt. Commerce students also have to study calculus. Like a commerce student, people who are planning to do economics, they have to study even in certain fields of maths. In certain fields of, you know, non-science subjects also calculus comes into picture. Right. So please try to understand how important is this for you, all of you. And in your class, I'll just give you a brief idea about calculus breakup for you. So in our class 11th and 12th, what happened to my board? Calculus overview. So I'll just tell you in calculus, you have to study, as I already told you, you have to study two branches of calculus. One is called differential calculus. Okay. And other is what we call as the integral calculus. Okay. I'm just talking from your class 11th and 12th point of view. In 11th, you will be basically talking about only differential calculus. Okay. Even though in physics, you will be talking about integral calculus also. But in maths, I'm talking about, you're going to start your differential calculus with the idea of limits. Okay. So today we are going to discuss limits with you. Then you're going to talk about differentiation. So differentiation is actually a process by which we find derivatives. Okay. So what is differentiation? What is derivative? All those things will be covered under this part. Now, limits is totally in your class 11th. Whereas differentiation is there in 11th as well as in 12th. More in 12th, less in 11th. 11th, you can actually treat it like an introduction. So you can treat it as an into in class 11th. Okay. In class 12th, you will be like going into details of it. Okay. Right now your 12th grade seniors have started with differentiation for them. So they are studying differentiation. In class 12th, you are going to be introduced to integral calculus. So under integral calculus, there are two types of integral calculus. We are going to talk about when it's called indefinite integration. Indefinite integration. And the other one is definite integration. Now, what is the difference between them? We will discuss when we start these concepts with you. In the bridge course only, I will try to talk about them if time permits. And both these topics are pretty challenging. I will not use the word difficult. It's challenging, right? Especially the first one, this guy. This is very, very practice intensive topic. I recall my class 12th summer vacation. So normally summer vacations are normally for 40 days, 40 odd days. For 40 days, I practice integrations, but still I was not confident. Still I kept on making mistakes, right? So it is very practice intensive because it is unconventional. Okay. Differentiation is very conventional. It has got some methods. It has got some standard operating procedures, which if you know, you will be able to solve almost 90% of the question. But integration is other way around. It is very, I can say it requires lateral thinking more than anything else. Okay. So these two topics will be taken in class 12. However, calculus doesn't end over here. You have application of integral calculus. You have application of integral calculus. Okay. Application of integration, what we call it. And you have only one application that is called area under curves. Now, how many of you, just I would like to know from you, how many of you are writing advanced placement calculus next year, May or this year, May. This year may probably most of you would not be able to write early 11th May. You will be able to write 11th May. Sorry. 12th May you will be able to write. How many of you are planning to write advanced placement? What is it? So if you're asking, okay. So advanced placement is basically, you know, advanced placements are exams which basically students write to increase the weightage of their application to foreign universities. So what happens? Let's say I want to pursue mathematics. Let's say University of California or University of Texas. So now when I'm filling this form, they will tell me what are the criteria I need to meet in order to pursue mathematics in University of Texas. So some of the universities say that you have to write AP calculus for it because AP calculus is a slightly advanced version of what you are going to study in 11th and 12th. So a person who has cleared AP calculus with let's say a score of four or a five, so total score is five actually. So then you have a better chance to get into that particular seat. Okay. So I think prism, prism, prism. I'm not getting your pronunciation right. So prism wants to write it. Anybody else? Okay. So you have to take this part very seriously application of integration. So you will have more components added to it. You will have to study volumes of revolution. There is a washer method. There's a disc method. All those stuff will come into picture. Okay. Now there is another chapter which I normally include in both of them integral and differential that is called differential equations. Are you sir equation is here also? Oh my God. I thought there were only algebraic equations. No my dear equations are not going to leave you. Okay. So you will be learning polynomial equations that you have already learned to a certain extent quadratic equation, et cetera. They are polynomial equations. Soon trigonometry will be introduced. You will learn trigonometric equations. Calculus is introduced. You will learn differential equations also. Right. So these are equations which involve no derivatives inside it. Okay. We'll talk about it when the right time comes. So differential calculus. This part is going to be there with you in your undergrad also. Let's say you get into IITs. Right. I went to IIT Kharagpur. And I remember my branch electrical engineering branch almost two years. I studied differential calculus only. So there are different types of differential equations. Sorry. Two years I studied differential equations only. So there are different types of differential equations that you will be coming your way. Okay. Linear, homogeneous, variable, separable, lot of things. Lot of things. Right. So this is going to be there for you if you are actually trying to pursue pure sciences or pure, I can say engineering later on also. Okay. So calculus is not going to end for you in 12. Right. So you are going to study more about it. So this is something which involves a mixture of both differential calculus as well as integral calculus. Right. So this session is primarily to, you know, apprise you of how important calculus is going to be. Of course, we are going to learn some concepts also. So now today, hi, Nikhil, I saw you. Nikhil was like, what is going on? Hello, Nikhil. So, yeah, so today we are going to begin with limits. Right. Because without limits, it's very difficult to understand differentiation. So limits is a precursor to differentiation. So let us understand limits today. Now, before I start a quick disclaimer, things may sound weird to you. Things may sound new to you, but keep your asking or where your asking skills or questions shoes on, keep asking questions because there are a lot of things which will be completely sounding weird to you. Okay. This is a disclaimer, which I'm giving you right now. It did to everybody who studied this course. It is not that you are studying it. It will sound weird to only you. I also whenever when I was in class 11, it all seemed what is happening here. What is this? But yes, later on I made sense of it because I asked right questions to my teachers and thankfully they were very, very helpful for me to understand this concept. Okay. So let's get started. So what is limit? Let me understand. Let me make this concept very easy for you. So first of all, I will write an expression in front of you f of X a function of X and to the left of it, I have written L I M and below it I have written X arrow a. Okay. So let me explain first of all, what have I written over here? So there is a function f of X that the user will give me. Okay. The question setter will give me the teacher will give me. And next to it I have written L I M. This is an abbreviation of, this is an abbreviation of limits a b v of the word limits. Many people will say that limit itself is a very small word, only seven alphabets, only six alphabets are there. Why you want to make it even smaller? That's a style. Okay. People don't want to write limits. Okay. Some people even do better. They write it as LT. That's another way of writing them. Okay. But what is important here is to understand this guy, this thing we need to understand. What does it actually mean? So this is to be read as this is to be read as X tending to a X tending to a or X approaching a. Okay. First of all understood how to pronounce it. What is it? I will explain you. So how do you pronounce this? X tending to a or X approaching a. Okay. Now let me give you this as an example to you. So if somebody says X is tending to two, what does it mean? It means that X is trying to come closer and closer and closer and closer and closer and closer to two, but never attaining to. Got it. What does it mean? X is trying to come closer and closer and closer and closer to two, but never attaining to or never achieving to. Are you getting my point? Okay. Now let's say, let's say I have a number line. Okay. On this number line. No, no, it's nothing like as input. I will talk about it on this number line. I have this number two sitting over it. Correct. So let's say a real number line. Okay. So since there's only one variable involved, I just took a single number line with for me and when I'm approaching to. Okay. Through this number line, I can approach this number to from two directions. I can approach to from this direction. Okay. Or I can approach to from this direction. Let me use a blue color. Correct. Yes or no. So when somebody, let's say there is a very small, you can say path. Okay. And you are standing somewhere in between. Let's say you are standing here in between. Okay. Now, you know that to your left in the right, there is, there are walls, right? Nobody can approach from there. So anybody can approach you either from this side or from this side. Correct. Yes or no. So let's say this is a, you are a. Okay. So you can be approached only from this side or from the other side, which is what now I'm making it in blue. So as to distinguish between the two sides, isn't it? So when you are approaching to from the left side, this is called the left side. Okay. So what all values can x take when you're approaching to from the left side. So x can take something like 1.9. It can take 1.99. It can take 1.999. It can take 1.999. There's no end to how many nines I can put. Correct. It can take 1.999. Billion number of times billion trillion, whatever you know, correct. So when you are approaching to from the left side of two, these are the values of x that x can take. Okay. So normally when we approach some number from the left side, we actually write it as x. Let me write it in white. So this is what we write as x tending to two minors. So the superscript, as you can see, basically says that you are approaching to from the left side of two. So minus just says you are slightly less than two. So minus doesn't mean minus two. It just is a representative or just is a symbolic expression that we use for conveying that I am approaching to from the left side of two. Okay. Yeah, you can start. See, Vaishna was asking, sir, can we start from negative side? Whatever you want to start with, it is basically where you are going. So I'm approaching to that is what I want to convey over here. Whether you start from 1.9, whether you start from zero, whether you start from minus hundred, whether you start from minus one lakh, it doesn't matter. What matters is you are coming towards to from this side of two, left side of two. That is what is matter of monitoring over here. Okay. So any question regarding this? I think Sharduli raised her hand. Sharduli, is there any question that you have here? Or is it addressed? You raised your hand a while back because of mistake. You don't know how to raise hand. Okay. Anyways, now if you approach to from the right side of two, then your ex can take values like 2.1, 2.01, 2.001, 2.0001, 2.0001. So basically what I'm trying to say is that you are coming close and close and close and close and close to two, but from the right side of two. And again, just to answer Vaishna, you can start from anywhere you want. You can come from infinity also doesn't matter. All it matters you are coming towards two. You are coming towards two from the right side of two. Is it clear? Okay. Welcome to calculus, right? Things are sounding weird because you've never heard of these kind of things before, right? Okay. Don't worry. Everything will make sense. But remember guys, when you are approaching to from the right side of two or from the left side of two, by the way, the expression for saying somebody that I'm approaching to from the right side of two is represented by x. Again, that tends to symbol two with a superscript of plus. Okay. So again, I like to repeat this. This minus doesn't mean minus two. Okay. This minus is just stating you that you are approaching to from a lesser value than two. So left of two. Two plus means you are approaching to from slightly higher value than two or a higher value than two. I will not use the word slightly here. So that is just a representation for you. Okay. Now, irrespective of whether you are approaching to from left or right, you are never attaining to. Right. So here note that X X will never attain to that means actually never become equal to two. It'll come very close to it, but never become equal to it. Are you getting this point? So 10 extending to two or extending to a is this concept clear to you. Any questions so far so good at least. Right. So whatever you have heard from me, it's not the same. Clear. Okay. So we'll go back to what is this expression? This whole expression actually. Okay. Now this whole expression means this whole expression means or let me write it like this. Let's say this quantity is L. Okay. Or let me write it as L I. Okay. Let me write L only. Okay. That you are evaluating is L. Correct. So basically this expression means this expression means this expression means as X starts approaching a f of X will start approaching or achieving L. Is this making sense? See here. There is a function which is dependent on X. So we all realize that when X will change, my function will also change. Isn't it? Isn't it? Isn't it correct? I just take a simple example. Let's say I take an example of limit X plus three as X tends to two. Okay. So do you all realize that as you change X here. X plus three will also get changed. That means your function will also change its value. Correct. Yes or no. So now I want to know. I want to know that when X tries to approach to where does this guy or to which value does X plus three tend to or to which value X plus three approaches to. That is what I'm trying to figure it out and that value to which it approaches is what we call as the L over here. Okay. The limit value. Are you getting my point? Right. Yes. Some of you have started giving the answer. Okay. We'll discuss it out. Don't worry. Let's go step by step. Okay. Now here one question which baffles most of the student is. So you are still approaching to whether you're approaching from the left side or whether you're approaching from the right side, your values of X are dynamic. Isn't it? So let's say I'm approaching to from the left side. So I take 1.9, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99, 1.99. So there's no ending to the 9s which I can add. Correct. So this X is not a fixed value. It is moving. It is a moving value. Correct. It's moving towards two. Correct. So whether you come from right side also, the same logic will hold two. So it is like 2.1, 2.01, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001, 2.001. So it is moving towards two. So when X is dynamic, it is moving. X plus 3 will also be moving. Correct. So this entire limits chapter is trying to address if X plus 3 is moving, where is it going towards? Are you getting my point? So where it is trying to approach? Or which value it is trying to approach? That value is the answer to this question. And that value is your L over N. Are you getting my point? Normally I give this funny example. All of you burn Diaz in your Diwali, right? Diwali time everybody lights Diaz or candle at home, right? Now let's say there is a Diaz. And let's say it is completely dark. You know when there is a light source, all these insects and all they will get attracted to it. I'm sure you would have experienced it also. So now let's say you figure out that there is an insect which has come very close to the Diaz. It is not taking a turn around it. Okay. It has not yet fallen into the flame. But basically here, when it is moving around, you can predict what is going to happen to that insect. Yes or no? Can you predict what is going to happen to that insect? You will say, yes, that insect is ultimately going to fall into that Diaz and Ramnam Satya. That insect will die, right? Yes or no? So you are predicting that it is going to approach its death. Even though at that point of time, it is still taking the circle around it. It is dancing around it, right? So at that particular time, you will not be able to comment anything. But you know for sure that after 2 hours, 3 hours or let's say in the morning when you get up, that insect would have fallen dead inside that candle or Diaz, whatever. Okay. So same way here also. X plus 3 is moving. But ultimately, it's going to attain some value and that value is what we call or it is going to tend towards some value and that value is what we call as the limit. Sharpen is saying RIP for insect. Okay. Now this may all sound a little bit weird to you, right? Because you are not dealing with fixed values. You are dealing with moving values. And that's the fun in calculus. Calculus is we learn about changing quantities, the quantities which are continuously changing. That's the fun. We are not talking about stagnant things. We are not talking about static things. Okay. So now let us evaluate it. So how can limit have a fixed value when value again? How does to have a fixed value? Right. So you asking this question is equivalent to you asking this question when X is tending to 2, why 2 is a fixed value? X is moving, but it is moving towards a fixed value. That fixed value for the function will be called as the limit. Or should I explain once again? See, X is moving, but it is moving towards a value which is fixed. Correct. Similarly, f of X is also moving. And sometimes it will be stagnant also. I'll tell you the cases. Don't worry about it. So f of X is also moving, but it is again moving towards the fixed value. Are you getting the point? Just like X is tending to 2, 2 is a fixed number. f of X is tending to L, L will also be a fixed number. Got it? Is that idea clear? Okay. So just like that insect, that insect is still moving, moving, moving. By the time you watch it, let's say you watch it for five hours, six hours, 100 hours, it is moving, moving. But you know that ultimately this fellow is going to die. Isn't it? So that dying is a fixed stage. There is no change in that stage. This L is a fixed stage that this function is ultimately going to achieve or approach. Is this fine? Now, these things initially they sound very, very weird to us. Okay. Even it sounded weird to me because, you know, some quantities moving and other quantities also moving, but still we say it is achieving or trying to go towards a fixed value. Okay. We'll figure it out. No. X is coming close to 2. It cannot achieve 2. For it to cross 2, it has to achieve 2. It will never achieve 2. See, let me just have a bit of patience. You'll understand a lot of things. Now see here. In order to evaluate this limit. Okay. So what is this answer? How do I find it out? Let's take an approach. Okay. By the way, this approach, we call it as informal approach. There's nothing in the name. Don't worry about the name. So I'm going to take an informal approach to find the value of L. Okay. All of you, please listen to this. So many of you would be thinking then what is a formal approach? There's a formal approach also, but I'm not going to tell you that because you're going to get confused. Okay. See, you are as good as in a person who is studying a particular thing for the first time. Okay. I don't want to tell you too many things because you will not be able to retain it. You'll get confused. Okay. So let's take baby steps. Okay. Let's take baby. Let's walk like a small baby step by step. Now X is approaching to here. So as I already told you, when you're approaching to, you could approach to either from the left side of two. And I also told you that we normally write it as extending to two with a super script of minus. Or you could approach to from the right side of two. So these are the two situations. So in order to distinguish between them, I have just put a line in between. Okay. Now, all of you, please contribute towards what I'm asking. If I am coming towards to from the left side of two, what all values of X I can take and what all values can X plus three take. That's my function. Okay. Now, let's take some dummy values. Let's say X becomes 1.9. Then what will X plus three become at that instant 1.9? I'm not asking you a rocket science question. 4.9. Correct. If I put 1.99, what does it become? You say 4.99. 1.999. You'll say 4.999. Okay. Now you tell me as you start going towards slowly, slowly, slowly towards two. Okay. But again, you are not approaching to. So we normally write it as two minus. Okay. Two minus. So two minus basically means you are in the very, very close proximity to two and to the left of it. Okay. So let me write this as a note for you so that you are able to remember this. When somebody says, when somebody says, I'll write it down over here. Note. I'll just write it down. When somebody says X is standing to a minus, you are basically in the, let me write it like this. Basically X is in the neighborhood of A and to the left of A. And to the left of A. Left of A. Okay. So normally, let's say I take the example of two minus. Two minus is 1.999. Keep on adding as many number of nines as you would like. So basically, when it becomes too big for us in mathematics, we cut short that big expression and we write two minus for it. Okay. So two minus is just a shortcut of saying it is like 1.999. On and on forever. So you can say 1.9 bar kind of a thing. Okay. So in maths, we don't write too many figures there. It looks stupid, of course, and we don't have time to write those. So in order to write that in a simple and straightforward notation, we say X is standing to two minus. So a minus is just a generic version of it. And when your X is coming very close to a. Okay. Let me write it as not tending to I will say equal to let's say. Okay. It means X is in the neighborhood of a by the way, neighborhood is an English word which says in close proximity just like your neighbors. Okay. They live very close proximity to you. So in maths also, we will use these words neighborhood and all even though it's an English word. So you are in the neighborhood of a and to the right of it and to the right of it. Okay. So when you start approaching when you start approaching values like 1.99999 in order to cut short that too many nines, we will just write two minus. Okay. Now you tell me where does this value approach. Where does this value approach. 5 minus exactly Shri Jani exactly Aryan exactly Prisham, okay, say two absolutely correct. Well done. Okay. So now you realize that you are coming very close to a number and that number is 5 right now let us do a similar activity let's do a similar activity here also when I am approaching towards 2 plus okay so here also I will take up some values of x okay just for our understanding let's say I take 2.1 then this value becomes 5.1 I take 2.01 this value becomes 5.01 I take 2.001 this becomes 5.001 if I take 2.001 it becomes 5.001 correct so you realize that slowly and slowly and slowly and slowly and slowly and slowly you are moving towards let's say 2.001 which again I cannot write but I will just write 2 plus let's say x becomes 2 plus then this value will become 5 plus correct yes or no yes or no now look at these two instances and tell me in both these cases x plus 3 is approaching which fixed number answer is loud and clear that fixed number is 5 that fixed number is 5 absolutely correct so this answer that you are going to write here is actually 5 got this point or not I am writing it as big for you to understand this okay are you getting this point right so please note x plus 3 is not achieving 5 neither in the left hand nor in the right hand it is coming very close to a number in both the cases that number is 5 okay now here is something which I would like to add to this discussion let us understand few more things so basically I'm just building on whatever we have done so far that's the beauty of bridge course in which course we start very slow we start absolutely assuming that you have no knowledge about anything and we keep building on those okay so in case you feel disconnected in any of the classes I'm not just talking about maths in physics chemistry please ask questions that's how we will understand that you have not understood it so Shardul has a question sir when you said 1.9 bar I tried to express in factions exact value of it comes out to be 2 oh yeah that's basically 1.9 bar you would have done this in class 10th also it comes out to be 2 actually but there's a difference between them okay anyways let's talk more about this concept so when you were trying to find out the value to which the function is trying to approach when you were approaching 2 from the left side of 2 right you got this 5 minus isn't it correct so we say that the left hand limit of x plus 3 as x tends to 2 minus is 5 in this case please note 5 minus is not a fixed number it is a dynamic figure as I told you 5 minus is 4.99999 written billion trillion zillion number of time so it is keep it is basically a moving number right till you add a nine another nine it is again a moving number but when you say about a limit limit is a fixed number so when you're trying to approach 5 minus we say that the left hand limit is equal to 5 are you getting this okay and here also when you're trying to approach 2 from the right side your your function x plus 3 is trying to approach 5 plus correct so we say right hand limit is equal to 5 so here we say right hand limit is equal to 5 are you getting my point right so limit is always a fixed number limit cannot be a moving number even though the function is moving that's the beauty here even though x plus 3 is moving either ways whether left hand or right hand but when you talk about a limit value whether you're talking about left hand or right hand or your answer itself it is always going to be a fixed value nobody can say limit itself is moving okay limit has to be a fixed value that's why it is called limit limiting case okay so now here understand this the limit answer that you wrote over here as 5 it is actually the same value that you got for your left hand limit and same value that you got for your right hand limit so here is a new concept that I would like to add over here limit is nothing but the same value that left hand limit takes and the same value that the right hand limit takes so unless until right hand and left hand limit values are equal you cannot find the limit out so limit is nothing but it is the value taken by the left hand limit and the right hand limit so when both are equal that value actually becomes your limit so now a question will come in your mind sir what if they were not equal then what will happen we will talk about that now even before that a question would have come in many of your you know many of you you would be thinking are you this 5 can only be obtained can be easily be obtained if I put x value as 2 here 2 plus 3 is 5 why sir you did so much of drama and all 1.9 1.99 1.999 and 2.1 2.001 and all okay so it may appear through this example that I simply took an approach which was not required I could have just put this 2 value over here and I would have got the answer as 5 right now that is not always true here clearly you got the answer by substituting but that is not always the case so I'm going to bust all these myths that you have by taking different different examples okay so two questions we have in our mind which is still unaddressed one question is what if my left hand limit right hand limit are not equal then what will be the limit in that case that's one question that is coming in your mind I'm sure most of you would be getting that question in your mind and second question that would be getting in your mind how is this answer different from finding f of 2 correct doesn't it look like putting you know value of xs2 and getting 5 so how is it different from that let us try to understand these two questions through examples so I will answer the second question before I answer the first one so how is how is this different from this so basically let me write this versus as if some match is going on right so this versus this so how are these two quantities different and when are these two quantities actually same right because in the previous example oh I'm so sorry in the previous example both the quantities were same just a second yeah in the previous quantities in the previous example both the quantities were same right right so are they same always or they can be different also let us try to understand this so for that I will take an example let's say I want to evaluate limit x square minus 4 by x minus 2 okay so this is my function and this limit I want to evaluate when x is tending to 2 when x is tending to 2 correct okay let's try to evaluate this now now the limit exists let me tell you that shardily so now the first thing that most of you who thought that f of a and limit of the function as x tends to a are equal you will get a shock here because the moment I put 2 in place of x I realize I yo it is giving me something which is undefined correct but let me assure you the limit of this function is a fixed answer I'm not going to tell that answer I'm going to evaluate it for you okay so here putting the value of 2 in this function did not work out failed correct right so I'll tell you very interesting story of my life when I was very small I was like around five years old okay uh I had a driver correct say to the answer is right whatever say so let's let's listen to the story I had a driver in my house who used to take me to school okay and no this driver was basically a very you can say expert in his skill so whenever he used to change the gear of the car I never used to notice that okay I always used to notice him doing this on the steering correct and one day I went to my dad and I said why have you hired this person in order to drive the car all he does is this even I can do it and I can you know go to school on my own then my dad said okay if you think this is how the car is to be driven he took me to the car so it was parked in the garage so he took it out and he asked me to sit there and he said okay you feel that the way to drive the car is just doing this on the steering try to do it so he actually put it to neutral and all I hope some of you know how to drive car and he started the car he put the ignition key on and I started doing this vigorously I don't know why the car is not moving that's how my driver used to drive it correct so basically what I saw is half the thing which he actually used to do right so there is something called pressing the clutch putting the gear on releasing it putting the accelerator on and then trying to steer the car through the steering so what I saw only was probably one-fifth the activity which he did correct so when you saw that example it gave you that half hazard knowledge that okay I can put a value in in my function and get my answer of the limit right but that was not the reality reality is something different the reality is much more deep so let us try to understand through this example so you putting the value did not work out it failed it is undefined but some of you are telling me that this answer is actually a four okay you are correct how does four come out we will discuss it so let us take our informal approach let us take our informal approach what was that informal approach finding the left hand limit right hand limit values separately so I'll take that informal approach again I'm not going to answer what is the formal approach because it is going to confuse you a bit so I'm not going to talk about okay so for you know informal approach again I'm going to take two scenarios one where I'm trying to approach two from the left side and another where I'm trying to approach two from the right side okay now let's take some values of x and let's figure out what does this guy achieves for those values of x same for this also so x value and what does this guy achieve okay now for this I would request you to have a a phone in your hand which has got calculator in it okay you may probably have a calculator already if you have it get it out okay now I'm not encouraging the use of calculator here it is just for you to solve this particular question so that we are fast okay now let's take some values which are less than two 1.9 let's start with 1.9 okay tell me when I put 1.9 what does this answer come out to be so all of you please take your calculator put 1.9 square minus four by 1.9 minus two what does it give you? Arnab says it gives 3.9 I'll believe you Arnab because you're correct okay yes or no everybody gets that everybody gets that okay now put 1.99 what do you get what do you get what do you get what do you get 3.99 thank you Arnab 1.999 1.999 3.999 very good now you must be appreciating and realizing over here that as my x becomes two minus this guy will become actually four minus am I right am I right am I right am I right Arnab I'll come to that enough okay I understand most of you are writing the factorization of x square minus four I'll come to that also there's definitely on my cards so if I ask you plain and simple what is your left hand limit what will your answer be what will your answer be what will you say four right don't say four minus okay four minus is the value to which the function is you know going towards but ultimately limit is a fixed value and that is four in this case okay is it fine any question by the way you should also apprise yourself of this notation which I'm going to use LHL it's a short form of saying left hand limit okay so it takes too much of time to write the whole thing down okay now let's come to the right side of 2.1 tell me what comes out if I put 2.1 square minus 4 by 2.1 minus 2 what does what does it give me you'll say 4.1 correct okay exactly 4.1 2.01 2.01 4.01 correct 2.001 guys you have calculators with you you need to be fast 4.001 correct so as I move towards 2 plus this guy will actually move towards 4 plus and in such cases we say the right hand limit becomes 4 is the right hand limit and left hand limit value equal are they equal yes very much they are equal and that becomes actually the limit value so this answer is going to be 4 are you getting this point are you getting this point now some of you were suggesting this to me why don't you factorize it okay I'll do that for you also so let us understand that approach as well by the way is this clear how I got 4 as my limit answer right so now in your school or in your j exam they may also ask you separately for left hand limit right hand limit also so there can be a question where they can just ask you what is the right hand limit or they can just ask you what is the left hand okay so you should be aware how to find that okay okay now let's come to what people were telling me on the chat box some people were saying why don't you factorize this guy as x minus 2 and x plus 2 okay and why don't you cancel this out and you are basically getting x plus 2 to deal with okay before I sir when solving do we have to find yes okay in this case or at least in the beginning part of your learning of limits you have to do that rearian later on we are going to cut short lot of you know unwanted stuffs and we will get the answer pretty quickly okay now here is a question all of you please pay attention on your screen I've written something very interesting I've written that this expression is actually x plus 2 is it correct is it correct very good I don't know very good Shardari means this expression is always x plus 2 sure sure do you remember which teacher used to say sure sure always in your class 10 can anybody name that teacher centre mechanic uncle sir yes you say sure every time okay yes my question is is this expression always x plus 2 no sir not on Wednesday Wednesday it is not expressed correct version right so this is a mistake which people do this is x plus 2 only when x is not equal to 2 my dear friend not always right when x becomes 2 you are not allowed to cancel 0 and 0 remember in maths you cannot cancel out zeros because if you do then somebody can say hey this is same as this let me cancel zeros and I can say 2 is equal to 3 this is not possible correct right math is very interesting you cannot bend the rules like this right so yes I agree with the people who say it is x plus 2 but my dear only when x is not equal to if your x becomes equal to 2 this expression will actually be undefined and that's precisely what happened when I tried to put when I tried to put 2 over here correct okay now when x is approaching 2 when x is approaching 2 anyways I do not have any business with this correct yes this is what most of you would be thinking so sir why are you worried about x equal to 2 whether it is defined undefined I don't need that I am approaching 2 means I am never achieving 2 you are absolutely correct so this is of no use to me this is useless to me okay right so basically you are trying to figure out you are trying to figure out where does let me draw the graph also for this let's understand it graphically so if I draw this function graph let's say this is my function graph okay so if I draw this function graph with respect to x it is going to be a straight line it's going to be a straight line like this with a small change in this there would be a hole in the graph a very small hole I will punch at x equal to 2 okay so at x equal to 2 there would be a small hole in the graph now why is that because as you can see in front of you I have written this function is undefined at 2 okay hence to show that undefined nature of the function I have to punch a hole there is it fine is this graph understood by everybody okay minus 2 I didn't get that enough what is what is why should I bother about minus two at minus two is everything is fine let it be zero denominator should never become zero that is my issue right okay so at 2 it becomes zero it is fine let it be correct but sorry at minus 2 it becomes zero here but at 2 it becomes undefined undefined is the problem for me zero is not a problem function can become zero no issues okay but you should not become undefined so undefined means I do not know what is happening to the function so I have to put a hole there because it is not known to me how is the function behaving gotilla so I put a hole there now see here when I'm approaching 2 I am approaching 2 either from the left side okay or from the right side when I'm approaching 2 from the left side let's say take 2 minus 2 minus okay probably will give me something here which is 4 minus okay so let's say this is 2 minus point okay and let's say this is I'll use another color 2 plus so at 2 plus okay this will give me 4 plus correct so both these values are trying to come close to a fixed number and that number is 4 and that 4 becomes your limit got the point okay now in the previous discussion I was avoiding telling you about formal definition right formal approach of finding the limit now I'll tell you sir what did you just do I just basically figured out that when I'm very close to 2 on the left of 2 my function is going to achieve 4 minus sorry the minus sign got hidden by that arrow and when I'm coming to 2 from the right side of 2 the function is approaching 4 plus so we say that in such situation your function is approaching a fixed value and that value is 4 which is precisely the limit okay so whatever is happening at 2 is none of my business whether there's a bomb blast at 2 I don't care whether there's an undefined happening at 2 I don't care okay I'm just concerned with what is happening before 2 what is happening after 2 and are those 2 values which I'm getting for the function they are basically approaching a number which is in this case number 4 and that 4 is your answer does it make a sense to you setu now as I was discussing with you I avoided something called the formal approach or the formal definition of limit now I'm going to tell you this okay because this graph is there for me to you know support it when you say when you say limit of a function as extends to a is a value let's say I write it as a value what should I write what should I write what should I write let's say I write an L just a second guys I think just a sec somehow I lost that this happens sometimes it happens so let's say if you call this number as an L okay then there is a definition which is called epsilon delta definition okay this definition says if you're saying that the limit of a function as extends to a is L that means when the distance or the gap between x and a is a quantity delta okay where delta is tending to zero plus or a very small quantity okay then the gap between the function and L will be a quantity epsilon where epsilon is also a very very small quantity now again let's try to understand this through this diagram so when this gap between when this gap between your x value let me let me just erase this part as of now okay let me erase this also okay now try to understand so it says that at whatever x you are let's say you are here okay and this is your two value so when this gap is very very small okay then the value of the function at that point and the actual limit of the function this gap this gap is what we call as epsilon that will also be very very small okay so it is just trying to say that it is just a formal way of saying that the gap between left hand limit and the right hand limit in this case four plus and four minus that gap will be almost tending to zero if that gap is tending to zero then the value to which these two left hand limit and the right hand limit are trying to go which is four in this case that will become your answer to the question okay now this people find it very difficult to understand especially a student coming from class tens right this is mostly used in undergraduate this is called the epsilon delta definition okay so please do not you know basically get this heartened if you don't understand this what you know should be your informal approach or what you should know is the informal approach so don't worry too much about it let me write it down don't worry too much about it okay because this is something which anyways you are going to see in undergraduate well you will pass out 12 we'll go to a college most of you would be in IITs there your professors will talk about epsilon delta definition of limits okay as of now don't worry about it okay this is the first class of your calculus I don't want to throw a lot of jargons at you okay so just understand that till your gap between till your gap between left hand limit and right hand limit if this gap is almost tending to a quantity which is very very small okay infinitesimal value okay this let me write it here infinitesimal value infinitesimal value right so that value of the left hand limit and or the right hand limit that would become your answer so in that case your limit will become what was your left hand or what was your right hand value both are basically going to be you know same so that is going to be the limit so that is what this formal definition says okay so don't worry about it now the next question which we did not answer was where is this gone now let me do one thing I don't know how to save it some some issue with this tool I know you can see this screen there's no problem with this screen here the problem is with this tool okay the problem is yeah now it has come yeah sometimes what happens no this this toolbar below and the toolbar on the right they disappear okay and because of that I'm not able to change the screen okay now I'm thank you thanks for the patience now here the second question is there a chance that the left hand limit and the right hand limit are different from each other right and in that case what happens to the limit so for that I would take a very unique function which probably you would have not heard of let me introduce you to something called the greatest integer function some of you might have heard of it some of you may not have heard of it so never mind for both of these group of people I'll explain this concept greatest integer function is basically a function which we represent by writing square brackets and writing something within it writing a function within it now if you see your graph and its transformation worksheets towards the lower part toward the end of the worksheet I've started using square brackets some of you would have thought those square brackets are just normal brackets right they're not actually there they stand for greatest integer function bracket okay now what this what does this function actually do this function returns greatest integer less than or equal to whatever you have put inside okay whatever function you have put inside for example for example let's say I put 3.8 so tell me the greatest integer which you feel is less than less than or equal to 3.8 of course there can be no integer equal to 3.8 but I can find an integer or greatest integer less than 3.8 and that has to be 3 correct vashna 3.799 is an integer for you okay yeah integer my dear integer what do I need I need an integer what type of integer there's so many integers below 3 below 3.8 right there is 3 2 1 0 minus 1 minus 2 but what integer I need the greatest integer less than or equal to 3.8 so that can only be 3 in this case what the point let's say 11.49 what do you think is the answer for this 11 very good now you have understood okay tell me uh 0.00001 what is the greatest integer 0 very good okay tell me minus 6.3 minus 6.3 uh now people have started making mistakes this is minus 7 I tricked you all okay now those who are saying minus 6 my dear minus 6 is greater than minus 6.3 minus 6 is greater than minus 6.3 but the definition says less than or equal to so you cannot cite anything more than that so the greatest integer which is less than minus 6.3 is minus 7 got the point okay now why I'm talking about this function all of a sudden because I would like to take an example on this let's say let's say I want to evaluate I want to evaluate limit of the greatest integer of x when x is approaching 3 or when x is tending to 3 by the way one small word before I move on whenever you see a square bracket please note that unless until stated that it is a greatest integer function please do not assume it to be because many a times square brackets are used just because there is a scarcity of brackets okay so whenever such a question will come you should always watch out for this word where this represents represents the greatest integer function gif okay the short form is gif not the ones which you send on your whatsapp and all okay that is also a gif that is called graphic image format but this is your greatest integer function okay so in your in your worksheet of graph and its transformation those brackets actually stand for gif even though I'm not mentioned it I should have mentioned it is my mistake okay so my point here is do not start assuming any square bracket to be gif okay so you should watch out for this particular phrase anyways tell me what should be the answer for this what should be the answer for this very good I'm getting answers some so many of you let's start working it out let's take our informal approach so for informal approach I would basically take these two scenarios extending to three from the left side extending to three from the right side okay now let's take x gif of x x gif of x okay we put dashes in now let us assume some values to the left of three let's start with 2.9 what is 2.9 gif two just now you did some questions with me okay what is 2.99 gif we'll say it says still two correct what is 2.99 gif we'll say it says still two correct so as you see if I go to three minus this answer will still be a two correct right so in this case remember when I was discussing the definition I said f of x may achieve also some value f of x may tend towards l but it may also achieve well also in this case it is actually achieving so here we can say that the left hand limit value is two correct now the game is not over yet we have to check out when it is approaching three from the right side of three so let's say I take 3.1 then what will your answer be for gif of 3.1 this time we'll say three 3.01 three again 3.001 three again if I go to three plus this will be three and three nothing else what did you see here and what did you realize here you realize here that for the first time the left hand limit value did not match with the right hand limit value correct so here your left hand limit value let me write it in red just to show that they're not matching so here left hand limit value did not match with right hand limit value then are you what should I do in this case so in this case my dear students we say that the limit does not exist so the answer to this question is limit does not exist okay in short form I will say dne for it does not exist okay so yes now welcome to another I can say perspective of limits limits may not exist also so it's not necessary that limit should be found out for every function for extending to anything okay so limit may not exist now any questions first of all let me let me just pause here for some time and give you an opportunity to ask this question we'll talk about it shardari that was next on my cards actually okay meanwhile here any questions you have okay now one thing I would like to add here please note that many people start blaming the function it is this function you know this this this this gif function now I'm not able to find limit no it is not that it is the whole situation to blame that means because of this whole question your limit did not exist it is not because of the gif function so don't put the blame on the gif function so I mean I don't want you to have uh you can say don't stereotype this don't be like okay whenever there's a gif their limit will not exist no it is not like that I'll give you another example with the very same function where the limit will exist let me give you another example so let's not stereotype things let's evaluate limit of gif as x tends to 2.5 okay let's see this so now again I'll make that I'll take that informal approach so I will approach 2.5 from the left side I will approach 2.5 from the right side okay so again let's take some values to the left and see what happens to gif and same I will do for the right as well okay now let's take some value tell me a value which is very close to 2.5 and to the left of it well you may start with 2.49 correct so 2.49 gif what is it my dear what is it what is it what is it 2 very good 2.499 this is a still 2 2.499999999 it is a 2 so as you approach 2.5 minus okay as you go to 2.5 minus this value will still be stuck at 2 correct yes or no now let's take 2.51 on the right side 2.5 plus part so 2.51 2 2.501 2 2.50012 so as you go towards 2.5 plus this will be stuck at 2 basically your left hand limit is 2 right hand limit is 2 and both of them are equal and both of them are equal so this answer is actually a 2 so here the limit is existing so don't try to stereotype this that okay this guy is a waste fellow I will not if that will never give me a limit no it is not like that it all depends on the situation getting my point now is this the only situation where the limit is said to not exist no there's one more situation we'll talk about it that is what Sharduli was pointing out let us say I asked you this question what is the limit of 1 by x minus 2 square as x tends to 2 okay let's say I take this question why is limit called limit because it is a limiting value of the function okay so that is a limiting value of the function okay now let's try to solve this question so when I take the informal approach okay so let me approach 2 from the left side of 2 and let me approach 2 from the right side of 2 again here let's take some values and try to figure out here also I'll take some values and try to figure out this okay now let's say 1.9 now this quantity if you calculate on a calculator I don't know how much it comes out to be it'll be probably 100 I believe something like that 100 what is it 100 correct yes right I hope I have not done a mistake in calculation okay when you take 1.99 it will be like maybe 10,000 right oh my god I've and I've reached the end of this scheme okay anyways if I go very very close to it this quantity will become very very large isn't it so as I go to let's say 2 minus this quantity will become very very large correct now infinity is a number which is very very big for us to express okay infinity is a number which is too big for us to express correct so this number is infinity so I have written a symbol for it if I take 2.01 I'll again get an 100 if I take 2.001 oh no sorry if I take a 2.1 I'll get a 100 if I take a 2.01 I will get a 10,000 okay and if I start going towards 2 plus this will again become a very very large quantity now here many people think sir this is same as this guy because you have written it as infinity infinity guys let me tell you this very straight away infinity is not a fixed number infinity is a dynamic number it is too big for us to represent that's why we use this symbol just because I've used this symbol at both the places doesn't mean they are same quantities right because they're not fixed both are moving so your left hand limit here and right hand limit here both are infinities okay both are very very large numbers but still I cannot write the answer of this as infinity we have to say it does not exist are you getting this point because infinity cannot be sorry limit cannot be in a number which is already a moving number as I told you limit is always a fixed value so it has to be a finite value are you getting my point so this is another situation where your limit will be set to not exist are you getting this point are you getting this point so it is not just because left hand limit right hand limit doesn't match it can also be because your left hand limit right hand limit both are becoming infinity or both are becoming let's say minus infinity then also the limit will not know set to exist okay and the example that you gave me sharduli in your example sharduli basically gave me a question this limit 1 by x minus 2 as x into 2 here if you see your left hand limit will actually become minus infinity right hand limit will actually become plus infinity and anyways they are not equal so this also limit will not exist okay so my example was a much more confusing one which people will get confused in so I started with this so to answer your question they are definitely not the same correct infinity minus infinity are infinitely apart okay so here left hand limit right hand limit are not equal but in this example it may appear they are equal so you may have a tendency of writing infinity please do not do that infinity can never be a limit okay so we say limit doesn't exist in this case so sharduli are you there are you listening me can hear me does it answer your question dear convinced okay good now before I move on I would leave you with a quick recap of whatever I have done and one small question so in today's class what did we learn of course we didn't do a lot of things because we didn't even touch upon differentiation it's actually a far first thing before we learn differentiation we have to learn limits so we first of all learn that if somebody says that there's a limit of a function as x tends to a is a value l it just means it just means that when x approaches a your function will approach l or at times it can achieve it okay I've shown you both the situations correct second thing we learned here was limit value should match with what you get for the left hand limit and what you get for the right hand limit okay and this value should also be a finite one okay that means if your left hand limit and right hand limit do not match with each other okay then then then we say that the limit does not exist okay or even if your left hand limit or right hand limit become infinity or minus infinity whatever then also we say then also we say limit does not exist okay so these are some things which we all need to keep in our mind okay now before we close this today's session which was a very interesting one I would leave you with I would like to ask you to answer this question of mine there is a function which behaves like this it comes from 2222 so this is basically y equal to two line okay all of a sudden at zero this function becomes undefined at this place and comes over here and again it resumes back okay so basically it is like a uh how do how do I compare this have you seen this heart monitoring machines they'll go up like this up like this up like this okay so this machine is like it is coming coming coming coming coming quick down and then it goes up like this correct so there's a quick you can say drop to zero and again it comes back so and then goes like that okay my question is what is the limit of this function as extends to zero let's see who answers this question and I'll also give options to you okay option number A it's two option number B it's zero option number C it does not exist and option number D none of the above okay let me put the poll on if you are able to answer this question I will think whatever I have explained to you you have understood it okay so please answer very responsibly else I will become very sad people have not understood okay hello I have got two answers both are different anybody who's not able to see the poll you can respond on the chat box okay so enough that last number you have written is that your answer okay enough why are you smiling so much I'm smiling because I can't believe people are giving different different answers that's why I'm smiling sometimes I smile when I'm sad also chalo janta 18 of you are there and only eight of you have answered it is just a touch and go question actually so you should take just five ten seconds to answer it okay in another 30 seconds I'll close the poll I would request you all to to please press on the poll button or at least write on the chat box if you feel you cannot see the poll okay last 15 seconds last 15 seconds last 15 seconds okay five four three two go come on five of you you're waiting for something please answer it's good enough time is given okay anyways 13 of you have answered this question out of which most of you have said b even though that most part is marginally most so three have said a six have said b and four of you have said c okay none of you went for d okay thank god none of you went for d now let's discuss this let's discuss this when you're evaluating a limit okay you worry about what is the value that the function is tending when you are slightly less than that number that is zero minus and slightly more than that number which is zero plus correct so if this function if I ask you if this function if I ask you what is the value of the function when you are approaching this function from the left side of zero what would your answer be what would your answer be what would answer be if you're approaching from the left side of zero let's say negative point one what is your answer to write absolutely sharduli and tell me what is the answer when you're approaching the function from right side of zero what will answer me what is the answer that is also two are they equal are they equal yes so your answer to this limit is two option number a is correct now didn't I tell you that I do not care what is happening at zero or what what is happening at a in this case your a is zero of course right whether there's a bomb blast whether it is undefined whether it is defined I care I don't want to use that word I don't care okay so you just have to watch what is happening just before and just after and are those values sufficiently close to each other or are they value equal to each other in this case they're exactly equal two and two so that value two will become your answer are you getting a point so most of you who said zero is because you were you were trying to basically or your concept was basically stuck at the point that limit is equal to the value at a no it is not they're different we have already discussed it in one of the examples are you getting a point okay so anyways never mind we will have more such opportunities by the way before we conclude this session okay I don't know whether we