 Okay, so limit is not a new concept for us. We have already done limits in our bridge course But that doesn't mean I will not cover the limits from basics Of course, I'll be slightly faster But please feel free to stop me if you want me to revisit any concept Okay, so what is limit? How is limit defined? So when you say limit of a function as X tends to a Okay, what is the meaning when you say this is some value L? What is the meaning of that? So we had discussed this in our bridge course that the meaning of this is you are approaching X You are approaching a but you are not Achieving a so this is to be read as X is tending to a or X is approaching a Approaching a okay, but never attaining a But never attaining a Never attaining the value of a Okay, so as X tends to a and it can tend to a from two sides of a which is what we say as the left hand limit Okay, this is called attaining coming towards a from the left side of it Just write it as left hand what left hand limit just say left hand of a And you can come towards a from the right hand side of a which is what we call as a plus This is to be read as right hand Of a Okay, so as you're approaching a or as X is approaching a whether from the left or whether from the right The value of the function f tries to attain a limit L or Attains a limit L Either it tries to attain or it tends towards L or it achieves L Then this value L is called the limit of the function Then this L is called the limit of the function This is called the limit of the function as X sends to it. Okay Now Remember a few things over here When I talk about limits When I talk about limits, there's something called left hand limit There's something called left hand limit Okay, this is called left hand limit and there's something called right hand limit Right, I'm sure you must have come across this while doing it in the school exams and also in the bridge course Okay. Now these values these values may be individually also asked or It may be asked as a combined question of limits So watch out for what the question is asking you is the question asking you left hand limit only is the question Asking you right hand limit only or is the question asking you the complete limit? Okay, so please read this read the question, you know and calculate your limits Okay, so this is called left hand limit and this is called the right hand limit as X sends to a Okay, now few important points to be noted Few important points to be noted number one limit of a function Is always a finite quantity Limit of a function is always a finite quantity. You cannot have limit as infinity. You cannot have limit as minus infinity However, however, your left hand limit and right hand limits could be However, your left hand limit and right hand limits could be infinity or minus infinity Okay, so these limits could be infinity or minus infinity okay so these limits could be infinity or minus infinity both okay so never ever write limit of a function as x tends to some value as infinity no what would you write if it is coming infinity you will write does not exist okay limit is always a finite quantity if limit is coming out to be if your left hand and right hand limit is coming out to be infinity do not by mistake also write infinity right does not exist that is a correct word to be used over here is this fine I thought you want to copy something here done copied okay okay so this is number one point number two number two let me go to the next page any question anybody see limit is a fixed value the word limit itself means it has a limit to it right you can't say limit is infinity for something then you're violating the basic meaning of a limit okay let me tell you just give me a second yes sorry there was some person at the door yeah so there is something called epsilon delta definition I'll go back to the previous slide shares once again there's something called epsilon delta definition of limits which probably we don't tell a you know grade 11th student because you will not be able to appreciate that but let me tell you that see let's say this is the graph of my function this is the graph of my function okay so there's something called epsilon delta definition of limit okay this definition says that if your difference between x and a is very small we which we call as delta delta is a quantity which is infinitesimally it is an infinitesimal quantity infinitesimal word is made basically made up of two words infinitely small infinitesimal I'll write it down for you infinitesimal infinitesimal is a combination of two words infinitely small so when your gap between let's say this is your a value right and you're trying to approach a whether from the left or the right okay let's say you are at a particular value x over here when this gap when this gap is infinitesimal then whatever is your actual limit whatever is your actual limit the gap between the value of the function let's say this is your actual limit I'm just taking a case of a continuous function but please do not get me wrong that you'll always get continuous functions to deal with so let's say this is your limit value okay then the gap between the function value let me show it like this so this is your function value and this is your limit so this gap will be another infinitesimal quantity which we call as dead epsilon which we call as epsilon so the gap between the function and the limit would be another quantity which is a very very small quantity which we call as epsilon okay so as you are approaching a this gap between the function and the actual limit value will also close will also become lesser and lesser okay so this L cannot be infinity that means you're trying to achieve infinity which is not possible right so in such cases when your limit is becoming infinity we will say it does not exist is that clear I think Shashot asked this question whose question was this yes limit itself will be a finite value that's what I wrote got it yes but left hand limit right hand limit can be infinity left hand limit and right hand limit can be infinity okay yes limit cannot be infinity but left hand limit right hand limit can be infinity so if somebody says if somebody says what is the limit of 1 by x as x tends to 0 minus right this answer would be minus infinity okay what is the limit of 1 by x as x tends to 0 plus this answer would be plus infinity okay but what is the limit of 1 by x as x tends to 0 does not exist this is how you'll say is that clear yes yes absolutely absolutely absolutely this is a definition which you'll see in your undergrad books okay in first year of engineering also you'll do a bit of calculus okay so now second property that I would like you all to note down just just make a note of the second thing if your left hand limit and the right hand limit values both are finite okay but they are not equal to each other okay but these two values are not equal to each other then also we say that the limit of a function as x tends to a does not exist okay this does not exist you'll come to learn later on that such kind of you know function which exhibit this property will be said to have a finite discontinuity at a but that is something which you do not bother right now it's a class 12th concept when the right time comes we'll talk about okay third point which is very important I would like to spend some time on it the difference between what is the difference between the value of the limit as x tends to a or the limit of a function as x tends to a and the value of the function at a what is the difference between these two now in order in order to in order for you to understand the difference I'll tell you some very interesting facts from those facts you will try to or you try to you know assimilate what is the actual difference between the function value at a and the limit of the function as x tends to a I'm not saying they are never the same I'm not saying that but try to understand the nuances between these two terms now three cases I will tell you number one you can have a function which does not exist you can have a function which does not exist at a that means f of a is undefined but you can still have the limit of a function as x tends to a existing okay I think this speaks volumes about the difference between the two so your limit can exist even if your function is undefined at that point any given a classic example for the same any give me an example for the same close to f of a you can't say close to f of a f of a might not exist this is an example f of a is undefined but still limit can be found out no on at all tan x tan x limit as x tends to 5 by 2 doesn't exist okay a very classic example for the same would be I'm sure you would have done this in your school as well let's say I have this function x square minus 4 by x minus 2 okay let's say I call this as my function okay this function you would realize that it is not defined at 2 f of 2 is undefined okay this is not defined at 2 but if you find the limit of this function as x tends to 2 you all know that the answer is 4 over a yes or no we all know that this answer is 4 okay so this is one of the biggest difference that you can see between the limit and the value of the function at that point okay number 2 your f of a may be defined or exists okay but the limit but the limit of the function at that point a may not exist can you give me an example for the same can you give me an example for the same let's see you give me an example signum function absolutely very good example signum function okay signum function if you take signum function I hope all of you know the definition of signum function I have told this and I know n number of times to people but still people keep on forgetting signum function is a function which returns one to you if your input to signum function is positive so it's like a you know we can say a method that you probably can write write course one python so let's say you you define a method okay def SGN x which gives you one when your input is positive which gives you minus one if your input is negative and which gives you zero if your input is zero so if you write such a method in python that method would be named as a signum function so it is giving you this function now we all know signum function graph signal function graph is like this okay now if you find if you know ask somebody what is the signum zero value signum zero value is zero but if you ask what is the limit of signum value or signum function as x tends to zero the answer for this will be it does not exist why does that exist simply because your right hand limit is one right it is because your right hand limit is one I'll write it down over here right hand limit will be one and your left hand limit would be zero minus one and your left hand limit will be minus one and they're not equal to each of them right another example another example where this can happen anybody gif function absolutely an Iraq gif function another example would be your gif function let's say I want to find the limit of gif function as x tends to one it doesn't exist it doesn't exist okay so let me just draw the graph for gif function gif function is like steps you've all seen this in the bitch course right so if I ask you what is the limit of gif function as x tends to one you will say it does not exist because your left hand limit is one sorry left hand limit is zero right hand limit is one that means you just look left or right of the function and just see whether are those values approaching towards the same value or are they the same value okay in this case neither is happening so left hand limit is zero right hand limit is one yeah sorry so that doesn't exist but what is the value of the function at one the value of the function at one is one okay this is another example okay third case third case which is a vital case just a second my dear students just give me a second somebody's at the and the final case is the one which is telling us is that yes they can be equal also they can be equal also so this can be equal to f of it okay and when does this happen when does this happen it happens in those cases please note down it happens in those cases where your function f of x is continuous continuous at x equal to it okay so yes there are occasions when the value of the function at that point matches with the limit of the function as x tends to a okay but this will happen only in those cases where function is continuous at x equal to it so what do we conclude from here if there is any kind of undefined nature shown by the function at a or if there is any kind of discontinuity shown by function at x equal to a then the values of the function at a and the limit of the function as x tends to a will not map will not be equal the only situation when they are equal is when your function when your function is showing continuity at x equal to it is this fine is this fine continuity means no break no break or no undefined characteristic at x equal to it please make a note of this please make a note of this very very important factor sir fraction part function as well yes of course fraction part is also an example for the second case you're talking about the second case right yeah definitely definitely so please make a note of this now I take up some questions just to test your basic understanding just to test your basic understanding whether whatever has been done so far whether you have understood it okay so I'm going to take some questions the first question that comes your way is this question sorry I'm looking at the wrong let's take this question which of the following statements are true for the function f defined from minus 1 to 3 shown in the figure okay let us look at the first one limit of f of x as x x tends to minus 1 plus as x tends to minus 1 plus the answer that they are given is 1 is this true just type it on your chat box is it true just like t if you think it is true a right f if you think it is false shares why do you think it is false shares shares would you like to yeah would you like to talk about why it is false no specific reason okay the answer for this is it is true see you just have to see you know little bit I mean you can say no very close neighborhood to the right of minus one so when you're very close to minus one the value is almost trying to approach one okay so this answer or this you know statement is actually true okay anybody who thinks it is false please seek for a clarification okay don't accept what others are saying I don't expect except what I am saying just debate over it just you know discuss over it second part limit of f of x as x tends to 2 does not exist write your answer as 1 and 2 else I will get confused that whether it was the answer for the first part of the second part so one is true for to what is the answer okay okay so this is the case where we say that the function is suffering from isolated point discontinuity so such discontinuities are called isolated point discontinuity now you don't have to know the name it is something which will anyway study in class 12 okay now when you're looking at the limit of this function as x tends to 2 you just see to the left of the function and to the right of the value sorry right of 2 and left of 2 what is happening at 2 is none of our business in limit we don't care about what is happening at 2 whether it is defined whether it is undefined whether it is defined at the very same place where it should be or whether it is defined little bit up or whether it is not defined I don't care I just see what is happening left and right of that point and are those two values approaching the same figure or are those two value exactly the same is what we are looking for in this case your left hand limit and right hand limit are exactly one so in this case your limit will actually be one and hence statement number 2 is false because it says doesn't exist what about third one what about third one limit of f of x as x tends to 1 minus is 2 okay good Pratik yes Pooja we basically we basically see what is happening just before and just after that point are the values at the same position or are they very close to each other that is what we see okay so those values are very far apart and very far apart means they're not equal I should not say very far apart because it's a it's a relative word actually then if the values are not equal to the left and the right of that point then the limit will not exist okay third one you're you're trying to find out what is the limit of the function as x tends to 1 minus means slightly less than 1 you are here and it is very close to 2 so you'll be approaching to so your answer is true because yes it is going to approach to is this fine okay fourth one limit of the function as x tends to 0 plus is equal to f of x limit as x tends to 0 minus true or false fourth one very good absolutely correct it is actually true both limits are both the limits will be the same and the limit value is actually zero in this case is this fine yeah Raghu Ram see slightly less than one your value is approaching to your values approaching to so this is your left hand limit at x equal to 1 that is what they are saying equal to 2 that is true what a value to sign to approach that is your left hand limit don't look at the exact value exact value when it is slightly less than 1 will be lesser than 2 it is trying to approach to okay let's take another set of questions let's take another set of questions okay let's take this one first one yes that is correctious first one what is the limit of g of t as x tends to 0 minus minus 1 very good very good I think all of you have got this right okay second one limit of g of t as x at t tends to 0 plus minus 2 very good what are the limit of g of t as t tends to 0 d and e very good very good okay not undefined does not exist does not exist yes fourth one limit of g of t as x tends to sorry as t tends to 2 minus 2 very good limit of g of t as t tends to 2 plus 0 very good so what is the answer for this does not exist okay my god all of you are so fast g of 2 g of 2 1 see here dark dot right that is the value at 2 actually that is the value at 2 okay limit of g of t as t tends to 4 4 absolutely absolutely so this is basically a continuous function so whatever value it hit it has at 4 the same value will be your limit as well remember the third point that we had discussed that will be the answer is this fine any questions any concerns okay now let us talk about let us talk about how do we actually evaluate the limit of a function as x tends to a okay so how do we find this out before we find this out we are going to talk about what type of questions will be getting so you will be given two types of functions primarily to evaluate the limit one which will be continuous at x equal to a okay and other which will I have already spoken about it I'm just repeating it which will be either undefined at a okay or has a discontinuity at a now discontinuity also incorporates undefined okay but I cannot just write one of them as it will create you know as it will give you half as our knowledge discontinuity is also in the function is undefined at that point correct but discontinuity could be because of other reasons also it could be like it is defined but the point is slightly above or below as I had shown you in the one of the questions that was called isolated point discontinuity there could be discontinuity because of the break in the function like your step function right so don't like treat them as exclusive of each other discontinuity can also incorporate undefined cases now when your function belongs to the former category where it is continuous at x equal to a then you don't have to worry at all your limit can easily be evaluated just by putting the value of a into the function but as I told you such cases will be very less okay your 99.99 percent of the cases would be coming under the second category so this type of questions will be very rare okay this type of questions will be very common so when your function becomes discontinuous at a because of the breakage or because of the missing point or without because of it is undefined etc etc then we categorize it under seven indeterminate forms so this will lead to indeterminate forms of the given question now indeterminate form I had already explained in the bridge course but I will talk about it once again what is an indeterminate form see let's say I talk about the same example x square minus four by x minus two this function this function is undefined at x equal to two correct we all know that this is undefined at x equal to two okay but who cares about what is happening at two because limit is only concerned with what is happening prior to two and post to two right so it is just talking about what is happening at two minus and two plus correct so when you are actually evaluating this case sorry let me write down the function here when you're actually evaluating the limit of this function as x tends to two remember x is not equal to two x is not equal to two here so when you're very close to two whether from the left side or whether from the right side both the numerator and denominator actually are quantities which are tending to zero okay such quantities in maths are called indeterminate forms okay I've already explained this to you in the bridge course so I don't want to spend too much time on this so how many indeterminate forms you will be coming across seven indeterminate forms will be coming across okay so let me name them zero by zero form okay remember even though I'm writing zero by zero please do not read it as zero by zero read it as read as tending to zero by tending to zero okay in your books and all they will write in order to you know keep it short and crispy they will write zero by zero form but when you are reading it you should always read like this tending to zero by tending to zero it's a healthy practice see exactly zero by exactly zero is undefined in maths okay that we cannot evaluate anything for that so it is not exactly zero by exactly zero it is actually tending to zero by tending to zero but the word tending to is sometimes skipped okay next is infinity by infinity form next is next is next is infinity minus infinity form next is I'll write it as zero into infinity but how will you read this how will you read this form this is to be read as tending to zero yes yes read it as tending to zero into infinity tending to zero into infinity okay next is one to the power infinity form now this surprises many people who think sir won't it always be one won't it always be one why are we actually studying this as an indeterminate form right its answer will always be one no it is actually not one it is actually tending to one it is actually tending to one to the power infinity and tending to one to the power infinity can take various answers starting from zero to infinity but always positive remember always positive we'll talk about it when we come to this form yes yes and it's tending to okay six one is we call it as a zero by zero we sorry we write it as a zero by zero but it is to be read as tending to zero to the power tending to zero tending to zero to the power tending to zero okay this this is the form that will be you know doing in our lopital forms so we'll be talking about it by the use of lopital and the seventh one i'll try to squeeze in over here seventh one is infinity to the power zero form but it is read as it is to be read as infinity to the power tending to zero tending to zero so this can also give you you know a finite answer okay yes yes whether you write zero into infinity or infinity into zero form it gives you the same thing it gives you the same thing no difference is this fine any questions any questions okay before we start solving any question i would like to run you through algebra algebra of limits okay so what are the rules under which limit work we'll talk about it because that will facilitate as to solve complicated questions the first tool that we'll talk about is limit of limit of some or difference of two functions is some or difference of the respective limits is some or difference of the respective limits okay now this looks like a very trivial you know very simple property but some complicated questions have been framed on this our property let me let me ask you a question let me ask you a question okay just think about it and then answer question is if limit of f of x plus g of x as x tends to a exists then which options which options can be correct which options can be correct okay option number a your f of x limit as x tends to a exists and your g of x as x tends to a exists okay second is your limit of f of x as x tends to a does not exist and your limit of g of x as x tends to a exists okay option c your limit of f of x as x tends to a exists and your limit of g of x as x tends to a does not exist and option d both the limits may not exist both the limit may not exist oh pretty long question but this is a very conceptual question most of the people in the first attempt they get this wrong okay multiple options are correct so think before you answer so more than one options may be correct okay i'm looking forward to your response okay aditya madhav chairs pranav anurag very good others feel free to answer let it be wrong doesn't matter doesn't matter okay very good arjun okay let's discuss this let's discuss this see obviously option a can be correct because if the limit f of x and g of x individually exists then it is quite possible that the limit of the sum will also exist okay so there is a possibility that if the limit of f of x but g of x exists both the limits f and g will also exist individually so option a is no doubt about option b now those people who are saying b uh try name them or would you like to answer on your so i think shashwat has said all the options hariharan has said all the options okay i think arjun has said b and c i would like you to give me an example where one of the functions limit doesn't exist and other function limits exist and the sum function also limit exists can you give me an example like that it is for everybody who has gone for b option so can you justify it if you're not able to find example can you justify me that yes there can be a case where one of the limit doesn't exist exactly one of the limit doesn't exist but the sum of the function limit will exist common sense also if you apply let's apply a common sense okay if your limit of f of x doesn't exist if your limit of f of x doesn't exist it can occur because of several factors it can occur because number one number one your left hand limit and right hand limit may not be equal even if both of them are finite that's one reason for limit not to exist okay or your limit is or basically your left hand limit and right hand limit are infinity right that may be the two reasons why the limit doesn't exist correct now if your limit of g of x exists and let's say i call this limit as let's say g okay then if you are evaluating the limit of the sum of these two functions okay so you have to separately evaluate their left hand limit and right hand limit correct so let's say left hand limit was l1 right hand limit for this let's say i call it as f1 because it's related to f function and this is let's say f2 let's say they're both are finite okay so if you evaluate its left hand limit what will your answer be let's say i take this case where it is both are finite and they're not equal so your answer will be f1 plus g and what will your right hand limit of the sum of these two functions will be what will be the right hand limit of the sum of these two functions that will be f2 plus g and they can't be equal because f1 and f2 are not equal correct even if when one of them is infinity or even if when both of them are infinity or minus infinity you can't say two infinities are equal correct so in either case my dear the limit if one of them doesn't exist exactly one of them doesn't exist it can never happen that there's some limit will exist i'm repeating this again if the limit of any one of f or any one of g either of the two okay i mean only one of them not both both will come later on right so if limit of f exists g doesn't exist or limit of g exists f doesn't exist then the limit of their sum will never exist so it's not only that you cannot think of an example there will never be an example like that so option b and hence option c cannot be correct so those who have replied with b and c either only b and c or in combination with b and c your answer will straight away be wrong okay what about d can d be correct can is there a possibility that you know f limit is non-existent g limit is not existent but when you add them their limit exists can it happen like that the answer to this is yes it can happen a simple example of it would be gif and fractional part right so if you take your gif to be your f of x okay we all know that limit of gif as x tends to one let's say i take an example of limit tending to one this doesn't exist okay and let's say i take my g of x as fractional part we all know that limit of fractional part also as x tends to one doesn't exist but when you add it but when you add it you end up getting actually x isn't it this we all know it's actually x okay and x limit will actually be one as x tends to one okay so despite both of them not existing your sum of the function limit may exist may may i'm not saying it'll always be the case okay that's why the question says which of them can be true okay so this can be true if you're if your limit of sum of f and g does exist then individually they may be non-existent are you getting my point here okay so this is very very important you know conceptual thing that most of us ignore while we are learning the stuffs okay rule number two oh mother has a question can it happen that both of them exist and the sum doesn't exist no no if both of them exist the limit of their sum will definitely exist okay okay next let us talk about the product if you're finding the limit of product of f of x and g of x remember we can find it out by individually multiplying their limits individually multiplying their limits okay now again i'm when i'm writing limit of f of x as x tends to a or limit of g of x as x tends to a that means i am ensuring that they exist okay so don't say that okay sir what if one is zero another is infinity no if it is infinity then you cannot use the word limit here okay you have to say it doesn't exist okay so i'm assuming that both of them are existing both of them are existing now here also one conceptual question can be framed okay so if you're if you're limit of f of x into g of x as x tends to a exists then which of the following options can be correct which of the following can be correct can be correct uh let me copy the options on the previous page because i don't want to sit and write them down again takes in an effort to write the four options what sir you're very lazy sir at least on teachers they should write no sir okay so let's let me copy this and technology is there use it okay let me go to the next page i have already crisscrossed the options okay i'll rewrite that part again that is good enough for me think carefully and then answer no need to be in a hurry to answer take one one one and a half minutes time and tell me which options which options can be correct take your time okay very good very good very good shashwat go demanchu not saying right or wrong to anybody i just want you to participate very good mehan mehabab sing share says very good okay arjun nice nice nice nice come on guys back up oh you wrote your answer okay okay sorry sorry i missed it out everybody please participate let it be wrong what will happen max to max i'll correct it right that is what we want right we want to learn okay the answer actually is all abcd is very surprising right answer is all of them can be correct okay first one is obviously correct first one i don't have to justify it if both of them exist then the product will also exist okay now let us talk about bc and d cases a typical example for bc and d cases i'll take sine x function okay let's say let's say i take sine x sine x is your f of x let's say okay or let me take one by x because option d says f of x limit doesn't exist okay so we all know that limit of one by x as x tends to zero does not exist okay let's take g of x as sine x okay we all know that limit of sine x as x tends to zero is zero is zero okay so one of them doesn't exist other other exist and that is equal to zero and when you combine it or when you multiply it that it becomes sine x by x we all know we have done this in the bridge course and i'm sure you would have done this in school this limit is known to be one okay and if you just interchange their positions let's say you call g of x as one by x and f of x as sine x then even your c will be correct so b and c will go together so if they can be both wrong or they will both be right so if b is right c will also have to be right because you can interchange your functions and do the same thing what about d can you give an example of a function where both the limit doesn't exist but the product limit will exist now probably you will not be able to think of an example you know right away but let us take this one this is your example for b and c i lied it over here for your notes now i'm giving you an example for d let us take a function like this let's take a function like this five when x is greater than equal to zero okay and two when x is less than zero we can all say here very convincingly that your limit of the function as x tends to zero doesn't exist because your left hand limit is two a right hand limit is five it doesn't exist and let's say i have another function which is two when x is greater than equal to zero and five when x is less than zero okay so here also limit doesn't exist here also limit doesn't exist because left hand limit is five and right hand limit is two correct now when you multiply these functions when you multiply f of x and g of x you would realize you'll always get a 10 whether you're looking at positive values of x or whether you're looking at negative values of x so you'll always get a 10 so the limit of this function as x tends to zero will always be 10 in this case okay so despite both the limits not existing there could be a case where the product limit exists absolutely for now your example is spot on very good okay so these are some conceptual tricks that you may fall into you know from understanding point of view so i hope this is clear so remember when f plus g exists either both of them should exist or both of them may not exist then also you will have f plus g existing okay but it cannot happen that exactly one of them exists but in case of product there is no restriction both of them can exist both of them can not exist only one of them will exist so even in all these three four cases you may have the product limit existing okay is this fine yes yes so i'm talking about these laws when both f and g exist exactly aditya okay next law next law if you want to evaluate the limit of f by g you can evaluate it by the use of this formula limit of f as x tends to a upon limit of g as x tends to a but here the restriction is your limit of g as x tends to a should not be zero okay are you getting my point so if the limit of the denominator becomes zero then remember your f of x limit as x tends to a should also give you zero then only the zero by zero indeterminate form will be holding and then you can use your further methods to solve it but of course not this method this method will fail okay try to get this you know very very important if you are getting your denominator limit that is limit of g as zero then your limit of your numerator must also be zero then only you can further apply some other concepts to solve the limit are you getting my point that means the problem is still on but if your limit of denominator is non zero and the limit of the numerator is non zero let me tell you in that case your limit doesn't exist are you trying to get what i'm what i'm trying to say see let's say i take this example limit of x tending to one x square minus one by x minus one if you try to solve it by the above rule of course you will not be able to solve it but what i'm trying to say is something different i'm trying to say that if you're evaluating the limit of the denominator you realize that this is giving me zero then this must also be zero then only you have a further scope of solving it right if you get a case like this let's say i give you a case like this x square plus one by x minus one and you try to solve it by this way then you would realize that your numerator limit as x tends to one will be two whereas your denominator limit as x tends to one will be zero so this guy will be zero and this is non zero in this case this limit will not exist does not exist right here the limit will exist here will here you can further apply your limit concept to solve it but of course this law will fail this law will fail but still there is a scope of it being solved so there's a scope of of it being solved but if you realize that oh my denominator is becoming zero and my numerator is giving me some fixed value then for such questions you can't do anything in that case you'll have to admit that the problem limit is non-existent is non-existent is this fine okay now what are those further methods that is what we are going to learn so when this strategy fails we are going to adopt further more strategies to solve it so we are going to pick up a heavier tool to get the job done okay let me erase this okay next property next property or next algebra I would say limit of f of x to the power g of x you can write this as limit of f of x extending to a whole raise to the power of limit of g of x as x tends to it I think we have done these properties in our bridge course also provided provided provided it should not be zero to the power zero form because if it is zero to the power zero form that means your base limit is coming zero exponent limit is coming zero and you decide to write your answer as zero to the power zero and by the way let me tell you zero to the power zero is not one it is undefined operation in max exactly zero to the power exactly zero is undefined operation okay so do not try to justify that something to the power zero is one so zero to the power zero will also be one no it doesn't work okay a non-zero quantity raised to the power zero it is one that is I agree to it okay so if it fails that means if zero to the power zero comes then this algebra will not work you have to go even beyond it you have to pick heavier tools to crack it okay so really you have to you know let's say ak-47 is not working then you have to probably pick up a tank okay like that what sir non-violent jokes no sorry violent jokes okay then we talked about composition of functions f of g of x f of g of x so f of g of x your limit can be found out by finding f of limit of g of x as extends to it but again this is provided so let's say I call this value as you know m let's say provided provided provided f of m should exist what do I mean by that f of m should exist let's say you had tan of ln x okay ln of some function right and you evaluated the limit inside and you got minus sorry and you got let's say pi by 2 or 3 pi by 2 then you know tan pi by 2 is not existing correct or tan 3 pi by 2 is not existing so in such cases your limit will not exist so this is provided whatever answer you get your function f should be happy with that input okay so your function f should be able to process that input okay for example you have log of something okay log of some function and you evaluated the functions limit and you realized you got minus 2 there you know log cannot work with a minus 2 argument log argument has to be greater than 0 okay so in that case you have to write your limit doesn't exist okay so some typical cases is yes exponential functions log functions they come under this category see there are many more rules I will not be you know able to go into all the rules okay for example sandwich theorem and all we'll talk about later on okay meanwhile another simple rule that I would like to discuss if you have limit of if you have limit of mod of a function okay then you can evaluate it by limit of mod of f of x okay provided provided this limit exists okay provided this limit exists and finally let's say if you have a function which is lesser than another function in the neighborhood of x equal to a in the neighborhood of x equal to a what are the meaning of neighborhood of x equal to a neighborhood is a word which is a phrase which is used very much in mass to describe an interval which is very close very close to a so here basically we talk about this interval this is called neighborhood of x equal to a where delta is a very very small positive quantity or you can call it as infinitesimal quantity so if function f is lesser than function g in the neighborhood of a then remember even the limit of the function f would be lesser than equal to the limit of function g as x tends to it is very obvious actually okay this is somehow related to your sandwich theorem which we'll talk later on not right now we'll talk about it later on sandwich theorem okay yes sandwich theorem also called squeeze theorem also called pinching theorem okay as of now I would like you to you know work out certain questions which is using these kind of basic rules so let me go with a question let's start with a question okay let's take this question oh sorry once again shashwat wants me to revisit the previous page sorry shashwat just one second yeah done shashwat shares has a question sir if you get indeterminate or undefined forms from application of any of these rules does it exist yes then these rules will fail there these rules have those have limitations okay so these are the basic rules which will make your life simple but in case the rule fails you may have to pick up a heavier tool to solve the question yeah Pranav I'm waiting only done okay yeah let's let's talk about this question let me put the poll on let me know if you are able to see the poll yeah poll is running okay very good so if you feel it is something else you can always go for option d yes yes mother if there's a composition we have to take the limit of the inside function okay 16 of you have responded I will close this in another 30 seconds okay five four three two one please vote please vote three of you are still left please vote very good very good very good two of you still left two of you two of you one one one is left one person is left okay all right so sharing the result with you almost equal votes have gone to b and d b and d let's check which is correct see let us evaluate let us evaluate the limit of the function as x tends to five by two minus minus now remember all these brackets here are gif brackets here you can read the question question categorically mentions that all the brackets all the square brackets are gif brackets correct so when you are slightly less than 90 degree can I say sine x will be slightly less than one correct yes or no and when it is slightly less than one gif of that will become a zero agreed yes is that fine okay now when you're slightly less than 90 degree when you're slightly less than 90 degree what happens to cos x cos x is very close to zero but still on the positive side right so it is very small positive number you can say so even that will become a zero yes from now it is gif okay oh yes you can add that the rule also mother I think mother reminded me of one more rule which I think I skipped but I'll just write it down thank you mother just rule number eight you can add that if you are multiplying a cons constant to a function and taking the limit then the constant just gets multiplied to the limit of the function okay yeah thank you for reminding with that mother okay let's go back to the question okay so here we have evaluated the left hand limit which comes out to be gif of one by three so basically it is gif of one by three gif of one by three is going to be a zero let's now find out limit of extending to pi by two plus now pi by two plus sine x will be still very close to one but slightly less than one right I hope you all can recall sine x graph this is 90 degree so slightly more or slightly less than pi by two you are just below one really you can say 0.99999 kind of a thing okay can I see gif of that will still be zero correct yes or no what about cos x when you see a graph of cos x let's see the graph of cos x cos x graph is like this when you're slightly more than pi by two means you are here this is your pi by two plus when you're slightly more pi more than pi by two remember you are at a very slightly negative value let's say negative 0.0000001 what is going to be the gif of that it's going to fall to minus one absolutely it's going to fall to minus one okay so it'll become this term but nevertheless if you simplify this if you simplify this you end up getting gif of two by three which is again a zero so irrespective of whether you are finding left hand limit right hand limit your answer is always zero that means your limit value is also equal to zero your limit value is also equal to zero which means option number a is correct option number a is correct let me see how many of you voted for a six of you only voted for it six of you voted for it good karthik very good okay now if I tweak this question slightly if I tweak this question slightly okay let's say I make another question out of it if I say limit of gif of gif of sine x minus gif of cos x plus one by three so this round brackets are normal brackets but square brackets are all gif okay so where this represents gif then what will be the answer then what will be the answer same or different yes mother was absolutely correct in this case your answer will be does not exist in this case your answer will be doesn't exist because for the simple fact that limit will not exist remember your left hand limit came out to be one by three right hand limit came out to be two by three so here your left hand limit and right hand limit will not be equal and if they're not equal limit will not exist if limit doesn't exist there is no point finding the gif of it see later on you learn something about composition of functions when you talk about f of g then for f of g to be existing g must exist are you getting my point if g itself is not existent then we forget about the entire function so if the inside function is not happy if the inside function is not defined then how can you you know work on an undefined value okay so in such case your answer will be does not exist again these are conceptual questions j will try to trick you most people will write zero for this are you getting my point so i'm trying to bring out this minute difference in your understanding are you getting my point so if limit itself doesn't exist what will you find gif of gif has to you know you have to have some value in this case your limit itself was having gif so you are saved the function itself was having gif so end result was the same so you got saved getting the point okay i would like to take one more conceptual question before i give you a break i know all of you would be earning for a break now sir stomach doing good good sir we want to eat something you are holding us back okay let's take this question good good mother happy to learn i'm very happy that you are all enjoying the classes try this one out i would like to see your response for this i'll put the poll on again square bracket is a gif square bracket is a gif radiance radiance when nothing is written and you're trying to feed that value to any trigonometric function you'll always have to put your calculators in fact calculator is not allowed but you have you always have to consider that in radiance mode three of you have responded so far okay let's let's wrap this up in another 45 seconds so those who are yet not yet you know put their response let's wrap this up in another 30 seconds from now not 45 anymore five four three two one please vote please vote five of you are still left kindly vote kindly vote kindly vote okay okay last person left to vote okay so again maximum people have gone for d okay so let's check whether d is correct or not now see if you look at the numerator term i'm sure you can factorize it isn't it you can factorize it as gif of x plus seven into gif of x plus eight correct you all can factorize it now let us evaluate the limit as x tends to minus seven from the left side of minus seven that means i'm evaluating the left hand limit as x tends to minus seven now remember gif and all have to be given a concrete shape so if you are slightly less than minus seven that's that means you are like let's say minus seven point zero zero zero zero one what will happen to gif it'll give you some integer that integer you have to place over here remember gif is always redefined as an integer are you getting my point so the outcome of a greatest integer function will always be an integer value which will be decided upon what is the interval where you're looking at that function so you're looking at this function at minus seven minus so gif will give you a minus eight do you all agree so if it gives you minus eight so your numerator will straight away become this divided by divided by a quantity which is tending to zero i'm i'm writing it like this for the sake of you know understanding and this will be tending to sign one now remember dear students this is of the form exactly zero by tending to zero and this is not an indeterminate form this is basically a case of just a zero nothing else nothing else people who think it is tending to zero by tending to zero no it is not it is exactly zero by tending to zero so what is the answer for exactly zero by tending to zero zero zero simple are you getting my point tending to zero by tending to zero had it been the case i would have gone into more analysis i would have taken a further approach to solve it but it is straight away right pure and simple zero by a very small quantity now no matter how small is that quantity if the numerator is zero exactly zero your answer will be zero okay so one day i asked the student what is limit of zero by h i started getting all types of answers sir one sir undefined right it is actually zero simple so this is a very small quantity it doesn't matter how small it is if the numerator is exactly zero straight away sorry is over there okay in the same way if i evaluate what is the limit of x tending to minus seven plus in minus seven plus you know that gif will give you a minus seven so your other term will become a zero actually it will become like this okay no term will still be tending to zero into till tending to sign one i can say tending to sign one tending to sign one okay so it will still be a straight away exactly zero by tending to zero case exactly zero by tending to zero case so this answer will again be zero okay so your limit will be zero your limit value will be zero and again most of janta got it wrong right this is where je will test you my dear students this is where je will test you exactly zero by exactly zero is undefined does not exist such a case see when will you get exactly zero will you get such a case when you're evaluating limit yeah you may get you may get right in such a case you just have to say the limit doesn't exist it's an undefined case okay the only scope of solving a limit is when you have either tending to zero by tending to zero or exactly zero by tending to zero we cannot solve tending to zero by exactly zero we cannot solve exactly zero by exactly zero so these these two cases cannot be solved getting my point so good now we can go for a break i think let me see how many people got this question right only three people got this right only three out of 26 of you i believe yeah we need to work on more questions okay so as of now please enjoy your break i'm taking the time of the break to be from 6 17 p.m right now let's meet at 6 32 p.m let's meet at 6 30 so now let's talk about what are the methods that we can use to find out the limits of functions as x tends to a so let's say methods to evaluate to evaluate limit of a function as x tends to a okay so i'll talk about few methods which are used in certain cases so depending upon the cases involved we'll be using these methods so let me tell you these methods are not like exclusive of each other it is not like you use one method only to solve a question there can be multiple methods involved to solve a given question okay so as for the situation you have at hand you will use these methods to solve the question so we'll talk about yes today i'll be talking about only zero by zero form anurag so the first method that we have is basically method of substitution method of substitution yeah sorry the small kids they are playing outside and they are screaming their lungs out okay method of substitution you must also be missing those days no evening time when you used to go out and play football you know everything is now center has spoiled everything okay method of substitution so method of substitution as we have already seen is use when you have when you have a continuous function at a so in this case you can use your substitution to evaluate the limits so only works when you realize your function is continuous at x equal to a now how would you realize the function is continuous at x equal to a do you have to make a graph for it to know it is continuous no you the first step that you should do when you see a problem is put the value of x into that function so a you try to put into the function if you realize that nothing wrong is happening nothing wrong means some finite values coming out from there then that value is your limit but having said that it is not as easy for example let's say if you have gif of x and x is sending to one and you put one there you get one so one is your answer no no so you have to be careful while you're doing that so if you realize that the function is changing the definition near you know the critical points is that one of the critical point is the is the value of x where you are evaluating the limit that time you should not be doing it if you realize that the function is becoming undefined that time you should not be doing it if you realize that the function is becoming infinity or the function is tending towards infinity when you are you know putting that value that also you should not do it so in those cases substitution will not work so i'm talking about only such cases which is very rare where you can use substitution method is where you find your function to be continuous at it a small example i would like to give not i would not take much of a time let's say i have three x square minus two x plus four okay you know it's a polynomial function will not have any break anywhere it will be continuous at all the points so if you put the value of x as one here x as one here then this becomes your final answer which is five i guess okay so five is going to be your limit now having said so please let me tell you that method of substitution might not be the first step in solving any question because such cases will be very very less of course your teachers won't want you to in a score mark just by substituting but remember method of substitution is always the last step of all limits question okay in the last step you basically use substitution to get your final answer next method is your method of factorization i'll be fast here because you have learned these methods in school as well as in the bridge course it is just a recap of it method of factorization works when you are evaluating the limit of f of x by g of x extending to a and you realize and you realize that putting a into the function will give you a zero that means f of a will give you a zero and g of a will give you a zero and also you realize number two that your x minus a can be taken out can be taken out or this is a factor of let me write it in this way this is a factor of f of x and g of x both okay in such cases what we can do is we can evaluate this limit by writing it like this into some other function let me call it as capital f of x this also some other function let me call it as capital g of x and then this is a problem creating factor remember i'm using the word problem creating factor because this was the one which is responsible for giving you zero in the numerator and zero in the denominator that means you're tending to zero by tending to zero form came because of this factor so once you eliminate that factor you are free to substitute okay provided you are you don't get another indeterminate form so provided this is of finite value okay if at all you get another indeterminate form then really you have to either repeat the process that means either some x minus a factors are still left in the function or you have to take a further route to solve it okay let us take problems because you already know the theory sir we are already aware of the theory let's have problems on them yes again if x minus a is there in the denominator left off and there is something like let's say x sin x minus a on the top then that will neutralize it so it depends upon the question that is there but yes if you have cancelled out all the x minus a factors and whatever left on the top is a non-zero quantity and below still you have x minus a then it is does not exist then the limit wins okay so let's have questions let's have questions let's have questions let's take this question curly brackets denote fractional part curly brackets denote fractional part remember you are evaluating the limit as x tends to 2 plus i'll put the poll on for this very good okay five four three two one go okay so most of you have given d as the right option okay remember you are evaluating the limit as x tends to 2 plus so only one sided limit you have to evaluate now how is the fractional part defined fractional part is defined as x minus gif of x correct so in this question if you are evaluating the limit as x tends to 2 plus can i say a fractional part could be written as x minus 2 correct because gif of 2 plus will be 2 so x minus 2 so basically your problem reduces to evaluating this limit okay i have broken x minus 2 square like this now as you can see this is a factor which was creating a trouble and i cancelled it out but as i think aditya asked this question what is this one more x minus 2 left that's why i chose this question then you realize that there is a sign term to take care of it now this is tending to 0 by tending to 0 form so this is tending to 0 by tending to 0 form okay and i'm assuming you have done a bit of limits in school also so what we can do is we can write x is equal to 2 plus h and write h tending to 0 plus so this will become limit h tending to 0 plus sign of h by h which we all know is going to be 1 so answer here is going to be option number c that is 1 any reason for why you know people wrote chose does not exist as their right option as their options it was not extending to 2 it was extending to 2 plus h thing okay h thing is basically used when you are trying to evaluate a limit where x is tending to a non-zero quantity now many people ask me sir why do you choose this approach okay because most of the standard limits that you would learn will be tending to 0 just just you know give a glance to your adi shalma or ncrt a formula list you would realize that almost all the limits that they talk about whether it is sin x by x tan x by x whether it is e to the power x minus 1 by x whether it is ln 1 plus x by x whether it is just few of them are not just for example x to the power n minus a to the power n by x minus a x is tending to a there but most of the limits otherwise will be tending to 0 so when we are learning those formulas we are able to connect to this faster okay so what i did was instead of x tending to 2 minus i took x as 2 plus h i took x as 2 plus x let me write again here x is equal to 2 plus h so when x tends to 2 plus your h will automatically start tending to a 0 plus correct so this entire problem will now get converted to this now why did i convert to this so that you people are able to relate to this faster right we all know sin x by x x tending to 0 plus or 0 minus whatever you call it that will be 1 okay so you are able to have a better connect with this formula that's why i chose that approach is this fine oh you're not learned that okay okay okay no problem we'll come to that haven't you done limits in your school none of you okay if you haven't done no problem we'll come back to that concept once again while solving the questions let's take meanwhile these questions it's almost towards the end i guess i've scanned through almost all the questions oh yeah there you go let's take this one oh so haven't you done oh ypr has not done oh okay okay okay what is going on in ypr which chapters are going on in ypr complex numbers p and c okay now here the problem that you see is actually not a zero by zero form right it is actually infinity minus infinity form but later on you will learn that these indeterminate forms are interconvertible right so one indeterminate form can be converted to the other zero by zero can be converted to infinity by infinity okay infinity minus infinity can be converted to zero by zero okay so while you're solving this question a simple hint would be to convert it to zero by zero form first let me put the poll on i think i forgot to put the poll this is a easy question so i would give you around one and a half minutes last 30 seconds five four three two one go please vote please vote eight of you are still supposed to vote make it fast okay time up most of you have gone with c 33% vote okay let's check see here first we need to first we need to take the lcm lcm as you would see here would just be one minus x the whole cube so you can take lcm as one minus x the whole cube which is factorizable as this okay limit extending to zero sorry extending to one so this will give you 1 plus x plus x square minus 3 so this is nothing but this is nothing but x square plus x minus 2 upon 1 minus x 1 plus x plus x square now x minus 1 or 1 minus x whatever you want to call it that is a problem creating factor that's a problem creating factor for us we want to get rid of that so let's see whether it is factorizable numerator of course you can see when you put x as one it becomes one plus one minus two denominator is definitely zero numerator is also zero so it is actually of the form zero by zero okay so let us factorize it so this is the factorization of the numerator part we have one minus x and we have one plus x plus x square now this and this will go off leaving a minus sign behind now put the value of x as one so it'll give you minus 3 upon 3 which is actually minus 1 so option b is correct chanta jinta was wrong people said c option b was correct is that fine i think somehow you messed up with this sign how's the factorization of a 1 minus x cube okay hariyaran is asking so a q minus b cube the factorization is a minus b a square plus ab plus b square correct so treat your a as 1 and treat your b as x so this is 1 minus x 1 square will be 1 ab will be x b square will be x square correct hariyaran okay good next question copy this if you want to excuse me sir yes sir sir is it possible to do this some by taking common denominator yeah i mean if you take a 1 minus x common out ultimately you'll factorize it only you know ultimately you will solve it and you know have 1 minus x in the denominator coming up you mean to say this right you made yes sir like this right yeah 1 minus x and 1 minus 3 thank you sir yeah we'll give you the same thing no no difference in the approach okay next question this time i'll start from the bottom of these okay let's take this one i'll give you three minutes for this time starts now poll is on i do not use lopital even if you're aware of it because uh that will defeat the purpose for your school exams in school exams you are not allowed to use lopital will anyhow learn it will anyhow learn it but let's not underestimate these methods because many times lopital may not be a very convenient way to solve a question almost two minutes yeah you have time don't worry you have around 45 seconds more last five seconds oh sorry last 15 seconds mishand said okay no problem okay five four three two one please vote please vote nine of you haven't voted yet please vote fast fast fast fast fast okay end of poll maximum janta has gone with b b for bombay okay let's check now the first thing that you should always do is substitute substitution is the number one step that you should always do while solving a question many people say sir i i'll get zero by zero i know sir so i need to suffer you know sometimes you may be tricked it may not be a zero by zero question it may be like two by three will come will come out as an answer right so that you'll end up wasting your time applying any other method so if you put one your numerator becomes six minus six that is zero denominator becomes six minus six that is also zero so the moment your numerator and denominator are becoming zero when you're putting when you're putting when you're putting x equal to one it means it means just a second yeah sorry so now if you put the value of one in both the numerator and denominator you realize you get a zero that means x minus one is a factor x minus one is a factor of both the polynomials so when you get x minus one is a factor of both the polynomials you would definitely like to bring x minus one out from both numerator and denominator that means you want to factor you want to factorize this as x minus one times something correct similarly you want to factorize the denominator polynomial which is x to the power six also as x minus one times something correct now what is this something in order to find this two factors you need to divide correct so you need to divide this polynomial that is four x cube minus x square plus two x minus five by x minus one right so for that most of you would have actually taken long division approach which I believe is slightly long right I could do the same thing by a shorter method which I call as the horner's method horner's method also called the synthetic division method synthetic division method okay so how does synthetic division method work let us understand that I'll take some time to explain you but the moment you understand it I think division will become very easy but first of all synthetic division method is only used when your divisor is linear when your divisor is a linear function what is the meaning of linear function that means here you're dividing by x minus one it is a linear function okay horner method will not work if your divisor is a quadratic or a cubic or any other polynomial whose degree is more than one okay so yes we did this in the bridge course right as you so this has a limitation of course if you are dividing by x minus one then this has an advantage also you will be able to do the process very fast without writing too much unlike you do it in case of long division so how does this method work let us understand this this method first we have to do some kind of sanitization work not the covid sanitization so here we have to first write down the dividend in decreasing powers of x without missing out any power thankfully it is already written in decreasing power of x and no power of x is missed that means cube square one and zero are all there then you borrow their coefficients four minus one two minus five so you just bring down their coefficients make this weird symbol any symbol you can make it is just to you know keep them together and separate it out from the divisor one now you're dividing by x minus one if you're dividing by x minus one keep a one here okay so two things number one you need to arrange your dividend in decreasing power of x without missing out any any power secondly you have to bring down the coefficients of this term that is four minus one two and minus five write the one over here okay why one because you're dividing by x minus one had you been dividing it by x minus two you'd have written a two had you been dividing by x minus ten you would have written a ten had you been dividing by x plus two you would have written a minus two there okay write a zero here write a zero here add these two so four plus zero is four now multiply these two these two you multiply okay you when you multiply one with the four you get a four write it over here okay then again add then again multiply one and three write it over here then again add then again multiply one and five write it over here you'll end up getting zero at the end this zero at the end signifies the remainder that we already knew because remainder had to be zero because it was a factor what about these terms what about these terms start writing them with from behind with power of x as zero then this will be three to the power three into x to the power one this is four into x to the power two added okay so basically this is your quotient this is your quotient okay so done so your answer for this will be this question mark here would be four x square minus three sorry plus three x plus five okay similarly let's do this division also now we are dividing now we are dividing x to the power six plus five x square minus two x minus four so let me write that down x to the power six plus five x square minus two x minus four now remember all the terms are not there so basically you need to sanitize it by writing it first like this x to the power six zero x to the power five because there has to be x to the power five term zero x to the power four zero x cube five x square minus two x minus four okay so this is what you need to do you have to ensure you are writing every power of x minus five x cube I you thank you for correcting that else it would have led to a lot of issues yeah thanks thanks so this will be this will be five this would be zero yeah anything else any any any other mistake in copying no that's fine okay now put down these coefficients one zero zero five zero minus two minus four again you're dividing by one so put a one here by default you'll write a zero here always this is always zero by default first term below the first term you'll always write a zero so one plus zero one one into one one zero plus one one one into one one zero plus one one one into one one six then this will become a six then six again then six then four four zero as you see last term is always zero I mean last term should be zero because you're dividing it by a factor right as if you're not dividing it by a factor then the last term would be a remainder okay now as I told you what is the quotient quotient start writing from behind four then six x then six x square then one x cube then one x to the power four then x to the power five fine so this will become I'll just correct it over here so this will become x to the power five x to the power four x to the power three six x square six x plus four okay now since I'm explaining you so much it is taking so much time but if you're doing it in the examination situation you can do it pretty fast right it can you can do it very fast okay now what should be the final answer so when you take the limit let me take my image on this side so limit extending to one x minus one four x square plus three x plus five upon this okay cancel out the problem creating factor these factor will go off now whatever is the remaining term you substitute remember what did I say method of substitution even though it may not be a very useful process to start with but you always end with a method of substitution so when you put one here it'll give you 4 plus 3 plus 5 which is 12 upon 1 plus 1 plus 1 plus 6 plus 6 plus 4 that's nothing but 12 upon 19 12 upon 19 option D is correct let me again check what was the answer oh Janta said B it does not be it is D for Delhi okay no problem let's take few more questions anything that you would like to ask here or you'd like me to explain once again do let me know okay next question I think we have done these questions okay try this one out yeah I'll put the poll on see it actually works on playing with the coefficient Anurag right normally when you do long division method you would realize that you are doing the same thing and horror method just takes the you know important thing and places it in form of numbers it's actually a method also used in Vedic mathematics okay so they don't talk about the variables at all because you are already taking care of it by you know arranging them in descending power of x without missing out any coefficient okay so it works on basically a pattern that you observe in your long division method so somebody observed that pattern and then he formulated a room and it became that synthetic division method you do one thing you try doing one problem by long division method and you try doing a method problem same problem by Horner's method and see what are the things that Horner method takes care of you'd realize that there's a pattern yes kind of have you done some Vedic mathematics course in junior classes okay oh really nice nice nice correct Aditya yes they're doing the same thing is just that now they're playing with the coefficients because the coefficient decides with which you what you're going to multiply isn't it okay last 30 seconds for this this is an easy question all of you should get this right okay five four three two one please vote please vote ten of you are still not voted okay end of fold so most of you have voted for D sorry C C 60 percent let's check see this is a case again of a zero by zero form of indeterminacy okay zero by zero form in fact I write it as zero by zero but read it as tending to zero by tending to zero okay so numerator I'm sure it is factorizable as x square plus 2x minus 2x plus 2 correct sorry x minus root 2x plus root even denominator if I'm not mistaken it is you can write it as 4x root 2 minus x root 2 minus 8 that's x plus 4 root 2 minus 1 minus root 2 in fact minus root 2x plus 4 root 2 so it is x minus root 2 upon x plus 4 root okay now since x is tending to root 2 the problem-creating factor is x minus root 2 so cancel it out rest is a case of substitution so this will become 2 plus 2 which is 4 this will become 2 root 2 this will become 5 root 2 so root 2 root 2 gone answer is 8 upon 5 which is option number C so how much time it takes not even one minute completely okay let's try this one limit x tending to pi by 4 1 minus cot cube x upon 2 minus cot x minus cot cube x are you sir trigonometry you are doing factorization methods all of us are in trigonometry okay actually it can be solved by method of factorization only that's why I gave you this question that's a hint actually for you try it out no options please post your answer on the chat box correct and that's a good approach cot cube it is cot cube in the denominator there's a cot cube okay and rock let's check how about others okay let's discuss it I think most of you have given the answer for it see the best way to solve this question is making an interim substitution interim substitution is the best way to solve this question do not do not go into trigonometric details because it is not here it is not using any kind of a trigonometric identity it's a dry trigonometric function where you can just manage with this substitution let's say I put cot x as a t so as x tends to pi by 4 t will tend to 1 correct so this will become limit t tending to 1 1 minus t cube by 2 minus t minus t cube okay now definitely when you're putting t as 1 both in the numerator and denominator you would realize that you will end up getting 0 0 and 0 by 0 so basically it is a 0 by 0 form again I'm writing it as 0 by 0 but it is tending to 0 by tending to 0 form numerator is factorizable let us factorize the denominator so in order to factorize the denominator I'm going to divide this by t minus 1 so what I'll do is I'll first write it down in order to use my Horner's method I'll write it like this okay copy down the coefficient which is minus 1 0 minus 1 2 you are dividing by t minus 1 so write a 1 over here okay then 0 minus 1 minus 1 minus 1 minus 2 minus 2 0 so you can factorize this as t minus 1 minus t square minus t minus 2 okay so your numerator will become your numerator will become 1 minus t upon 1 plus t plus t square denominator will become 1 minus t t square plus t plus 2 remember I just flip the position of 1 and t and I just absorb the negative sign now this is the problem creating factor remove it and whatever is left off you can use your substitution method so it's 1 plus 1 plus 1 by 1 plus 1 plus 2 3 by 4 absolutely correct let me see who all got 3 by 4 no Kaushik Anurag that's correct Aditya that's correct Pranav got it correct Madhav got it correct Hariharan got it correct Aditya got it correct nice very good very good very good let's move on to the next question oh let's move on to the next question evaluate evaluate and limit okay let me give it like this limit extending to a mod x cube minus 6x square plus 11x minus 6 by x cube minus 6x square plus 11x minus 6 does not exist then then the number of solutions of a the number of solutions of a is option a 3 option b 2 option d 1 sorry sir you forgot ABCD sir what happened option d 4 if this limit as extends to a does not exist then how many possible a's are there that is what the question is asking how many a's are possible should I run a poll for this okay let's have a poll for this yes Anurag you're almost on the right track so that should make it very obvious five of you have voted as in I didn't get you what what you're missing you're on the right track absolutely yes correct okay last 20 seconds to wrap this 14 of you haven't voted yet okay five four three two one go yeah we'll discuss it we'll discuss it we'll discuss it don't worry Anurag we'll discuss it okay so most of you have voted for option b option b which is 2 okay now let's look into this if you look at the term x cube minus 6x square plus 11x minus 6 this is a very familiar term I'm sure you must have come across this term before it is actually factorizable like this yes now if you make a wavy curve for the same if you make the wavy curve for the same you have plus minus plus minus so these values signify the sign of that expression in different intervals of x correct now the critical points here are 1 2 and 3 now if you're evaluating the limit at these points a problem will arise why I'm saying so okay let's say I take one if I'm evaluating the limit as x tends to 1 okay left hand limit at 1 the function is negative correct so if I evaluate at 1 minus the numerator part would become negative x cube minus 6x plus 11x minus 6 okay and as a result your answer would be minus 1 correct but when you are evaluating at 1 plus the numerator part is a positive expression so it will remain as it is and then your answer will become positive 1 sorry positive 1 so left hand limit and right hand limit will be different from each other and hence the limit does not exist okay so one can be an example of such a value of a where the limit will not exist okay now 2 can also be an example like that 3 will also be an example in 2 the left hand limit will be 1 right hand limit will be minus 1 correct in 3 left hand limit will be minus 1 right hand limit will be plus 1 so at 1 2 and 3 the limit will not exist for sure correct now how am I sure that it is going to exist for rest other points see if I take a point let's say if I take a point 0.9 also okay let's say I take a point 9 what is the left hand limit at 0.9 and what is the right hand limit at 0.9 both will be minus 1 are you getting my point so when you are tending towards 0.9 from left side of 0.9 or right side of 0.9 the limit will always be a minus 1 so limit will exist at 0.9 correct take any other point let's say you take 1.3 it will exist take 2.7 it'll exist take 3.4 it'll exist okay only at these points problem is going to arise so the number of solutions of A will be 3 are you getting a point okay so only for 3 values of A which is 1 2 and 3 there would be no limit existing and hence your option is option A because they ask you number of solutions okay yes it is actually you can say 66 percent signal function plus 1 and minus 1 of course 0 is not going to be coming up from there okay so thank you next class would be a very important class we'll be talking about rational method of rationalization and other standard methods like exponential logarithmic algebraic all those methods of evaluating the limit okay thank you so much and all the best for your coming UTS take care bye