 Welcome to the first session on Limits. Limits is not a new concept to you because we have done extensive limits in our bridge course. But that doesn't mean I will not talk about the basics. Of course, I'll start with basics, but you can expect me to be slightly faster. But at the same time, if you feel you want to revisit a particular concept again, please feel free to stop me at any place you want me. So good evening everyone. Welcome to the first session on Limits. In the bridge course, we had already discussed the informal approach of finding the limit. So in informal approach, we had taken some example and we had understood how do we find out the left hand limit, right hand limit by trying to come close to the value that X is tending to from left and right of that value. And then we see what is the value to which your function is tending to. So when I say limit of a function as X sends to A is L, it basically means that it basically means that as you're approaching A, whether you approach from the left of A, whether you approach from the left side of a left hand, or you approach from right hand. Okay, basically left hand is what we write as extending to a minus. Okay, so on the superscript we put a minus sign that doesn't mean minus a again, I'm repeating it a with a superscript of minus doesn't mean minus a it just means you are approaching a from a slightly lesser value than a. For example, if A is four, you are approaching a like 3.9, 3.99, 3.999 such approaches towards four from slightly lesser than four is called approaching four from left hand side. Right hand limit is what we write as extending to a plus. Okay, so as X tends to a this is what we read it as X tends to a the value of the function tends to L or may even achieve L. Okay, then this value is what we call as the limit of the function. This is called the limit of the function as X sends to it. And this value L should be the same irrespective of whether you are approaching a from left to a or right to it. Okay, this is the basic informal definition that we had informal definition of limits. Okay, there is something called epsilon delta definition, which is not a very great significance to us. But since many books talk about it, we'll also talk about epsilon delta definition, epsilon delta definition definition. It says that if the difference between X and A is tending to delta where delta is a very, very small quantity, okay, then the difference between the function and L should tend to epsilon. Okay, where delta and epsilon both are infinitismally small quantity. So where delta and epsilon are infinitismal quantities. Okay, now unlike the bridge course, I will not be going into again, some examples. You have already done this in school. So we'll talk directly about methods to evaluate limit. Okay, anybody has any confusion about the meaning of limits? Yeah, which definition should that you want me to repeat epsilon delta definition, epsilon delta definition is basically when your gap between X and A is tending to zero, which we call as delta normally, then the gap between the function and the actual limit of the function as X sends to a will also tend to a very infinitismal quantity, which we call as epsilon. Okay, so this is the epsilon delta definition. If you want, I can show a small graph for the same. So basically it tries to say that, let's say there is any function like this. And this is your a value. Okay, so let's say you're very close to a very close to a either to the left of a or to the right of a, right. And this gap is your delta gap. Okay, then your gap between the value of the function in the neighborhood of L, this gap is what we call as the epsilon gap. Okay, so as you're closing the gap with a, even the gap of the function and its limit is closing its gap. Okay, so f will approach L as X will approach a, that is what I'm trying to say in a different way that is called the epsilon delta definition. Okay, so this is what is normally seen in many books. That's why I thought I will talk about it. But this is not of much use to us. Well, in undergrad, you'll talk more about it. Okay, now what are the, what are the standard ways or what are the methods to solve limits before we go on to that, I would like to highlight few important things. Very, very important pointers about limit. Okay, I'll also check your understanding of your limits by giving you some questions. But before we proceed, I would like to discuss this number one, number one, limit of a function as X sense to a is always a finite quantity. It's always a finite quantity. Remember limit, you cannot say limit is infinity. Okay, however, however, you can say that your left hand limit or your right hand limit, your left hand limit or your right hand limit could be plus or minus infinity. So these guys could be plus or minus infinity. But under no situation, you will say limit is infinity. What will you say instead? limit does not exist. Are you getting my point? But it may happen that your left hand limit is infinity, or your right hand limit is also infinity, or your left hand limit is minus infinity, right hand limit is also minus infinity. So please remember infinity and two infinities are not the same. Okay, so in those cases, you will say limit does not exist. Right, even if it is plus infinity minus infinity, then definitely it will not exist. That is number one point. Number two point is number two point is if your left hand limit and right hand limit values, they don't match despite being finite. Let's say these two quantities both are finite, but their values do not match. So despite being finite quantities, their values do not match. Then we'll say in such cases also, that the limit of the function as x since to a does not exist. Okay, this does not exist. D and E. Okay, even if there are those are infinite. So even if let's say your f of x as x since to a plus is tending to infinity, and same is your limit as x since to a minus, even in this case, you will say, even in this case, you will say that this does not exist. Okay, this I've already talked about a few minutes ago. I don't want to repeat it again. Third important point which I would like to discuss with you in some detail is the comparison between the limit of a function as x since to a and the value of the function at x at x equal to a. Okay, what are the difference between these terms? Are you experiencing any kind of a lag when I'm writing? Anybody's experiencing any kind of a lag? No, it's fine. Absolutely fine, right? Okay. Now I would like to discuss this difference unnoticeable. Okay, I will discuss this difference in some detail with you. Of course, they are different. But at the same time, at certain conditions, they are same also. Okay, so I would like to discuss the similarity as well as difference between these two concepts. Okay, of course, they are not the same always. Else, I would not have studied or you would not have studied this chapter separately. You would have studied it under functions chapter only. Okay, right? Yes, that is what I'm going to discuss. Okay, so I would like to discuss you cases with you. Okay, the first case is a case where it may happen that that limit of a function as x tends to a exists, but the value of the function at a does not exist. Okay, so what I'm trying to bring out, I'm trying to bring out certain things which will make you realize that they are not the same or they are same when they are same. Okay, so the first point or first case which I'm going to discuss is there may be cases where your function limit as x tends to a will exist, but the value of the function at a may not exist or does not exist. Can you give me an example like this? I'm sure you would have done several examples where this particular case has been or whether this case is seen. Right? So one example I would cite here is let's say limit x square minus 4 by x minus 2 as x tends to 2. Correct? Correct? Okay. No, no, no, no, no. When people are citing the wrong examples now, green function, Raghav is defined for all values of X. There is no such value of X for the signal function does not exist. You can find signal of any value. okay yes sine x by x is a good example sine x by x doesn't exist at zero but the limit as x into one sorry x into zero becomes one okay yes there are several examples so this limit we have all seen that this limit is four correct but if you talk as a function here this function so let's say if you have this function x square minus four by x minus two okay f of two does not exist it is not defined because your denominator of course numerator also become zero so it is exactly zero by exactly zero means the expression is undefined okay but limit of x square minus four by x minus two as x tends to two exists and it is four correct this is number one this is case one which I wanted to show you now look at case two look at case two it may happen that limit of the function as x tends to a what happened to my equal yeah x tends to a does not exist does not exist but your function at a exists okay right this is a second line of difference that you may see and can you give me an example for the same can you give an example for the same now signum example that he cited that's a beautiful example for this so if you want to find out what is the limit of signum function SGN function as x tends to zero this does not exist why does it it why doesn't it exist it's just that simple if you draw a signum function graph I hope you all know the signum function graph signum function gives you zero when x is zero gives you one when x is positive and gives you minus one when x is negative right yes or no so the definition of signum function I'll write once again many people asked me sir what is the definition of signum function I still don't know it so it is one when x is greater than zero it is zero when x is equal to zero and it is minus one when x is there than zero as you can see if the left hand limit the left hand limit of the signum function as x tends to zero is minus one the right hand limit as x tends to zero plus is plus one and these two values are not the same so we say limit does not exist limit of signum function as x tends to zero does not exist okay but what is the value of signum zero that's zero that exists is this fine is this clear yes mod x x can also be taken as an example can shoot very good yeah are you getting up on any questions now the third is what we want to discuss with you that they may be equal also third case is where your limit of the function as x tends to a and the value of the function at a may be equal both may exist and both are equal when will this happen when will this happen this happens when this happens when this occurs when your f of x is a continuous function at x equal to a okay a simple example for the same I would I'm sure I've given such examples before to you let's say x plus one as x tends to one okay this answer is two and even if you find out what is the value of sorry what is the value of this function f of x defined by x plus one at x equal to one that will also be two so both these values are equal and it is because such a function is a continuous function at two there is no problem okay so x plus one graph is like this so at x equal to one if you see the value of the limit and the value of the function both will match with each other okay so from these three cases you realize that they are not the same things f of x at x equal to a and limit of f of x as x tends to a they are not the same things I hope you can realize when does the difference come in the difference comes in when your function becomes undefined at that point or there is a discontinuity occurring at that point is this fine so please do not interchangeably use these two concepts fine now I will I'll give you some questions to begin with because I want to test your basic understanding let's start with let's start with a very simple question I have this question in front of me let me go to the next page I hope you can all see the question I'll just zoom in a bit yeah there you go okay for the function g whose graph is given state the value of each quantity if it exists state the value of each quantity if it exists okay now I would like you to answer the first part what is g of t as x tends to zero minus very good almost everybody is giving the right answer minus one good well done second one what is the value of g t as x tends to zero plus kinship it's minus two yes excellent very good guys third what is the limit of g of t as x tends to zero very good non-existent dne very good dne absolutely see this is the basic things let me tell you guys Jay will actually test you on these basic things don't expect very difficult questions to come in but they will just try to attack you know the cons you're understanding through these simple problems okay limit of g of t as x tends to two minus very good to absolutely if anybody feels that he is not agreeing with my answer raise your hand speak out put your concern on the chat box I'll answer your concern okay g of t as x tends to two plus zero very good very good g of t as x tends to two dne very good very good next seventh one g of two g of two g of two one absolutely as you can see there's a dark dot over here g of two is one very good eighth one g of t as x tends to four that is four yes because as you can see slightly left to four it is tending towards four slightly right to four it is also tending towards four so I think the first the primary tests that I wanted to give you you have all cleared it very good your understanding about that is very good so let's take another question I hope you can all see the question yeah for the function f whose graph is given state the value of each quantity if it exists okay what are the limit of f of x as x tends to one what are the limit of f of x as x tends to one very good Shadda why was a mistake for as x tends to one see left to limit left to one you are near to two right to one you are near to three the value doesn't match so your left hand limit and right hand limit do not match your answer is dne doesn't exist but so those are making a mistake now it is clear why the mistake happened okay tell me the second one limit of f of x as x tends to two three very good excellent third one limit of a function as x tends to three it's three only see what are you doing the mistake that you're doing you're looking at the value of the function at three okay limit and the function value need not be the same it is same and they're continuous here obviously there is a discontinuity at three right there is a this type of discontinuity later on you learn is called isolated point discontinuity okay so left to three sorry for writing with white in white left to three the limit is three right to three also the limit is three are they equal yes three will be the answer okay fourth one limit of f of x as x tends to one point nine nine fourth one if if possible please put the question number limit of f of x as x tends to one point nine nine the answer is three only okay now see one point nine nine many people think it is a quantity which is very close to two no one point nine nine minus and one point nine nine plus it is different from saying two minus and two plus are you getting my point so one point nine nine let's say it is a point like this okay you just have to see what is happening before one point nine nine and one point and after one point nine nine in this example before one point nine nine and after one point nine nine the values are both at three so limit here is three are you getting my point so don't treat one point nine nine to be the same as a two okay there's a lot of difference next what is the limit of the function as x tends to three minus three minus just before three that is also three so I think all these values are going to be three okay so this part is clear no mistake in this very good now let us talk about what are the methods that will be following what are the methods that will follow to evaluate the limit of a function as x tends to a okay I'll switch off my camera for some timing because I feel there is a small lag when I'm writing it just give me a second if it works yeah now when you're evaluating the limit of a function as x tends to a the approach that we take depends upon what kind of function are we dealing with okay so normally we deal with two types of functions so this function f of x can be of two types one which is continuous at x equal to a one which is continuous at x equal to a for such cases you don't have to worry because the limit of this function as x tends to a will just be equal to f of a okay I think the third point that we had discussed today it is in line with that okay the problem arises when your function is discontinuous at x equal to a or undefined or undefined at x equal to a okay so when a function becomes discontinuous at x equal to a or it becomes undefined at x equal to a then we basically have to apply our methods that we are going to see in some time now under these two cases under these two cases we basically call the function as an indeterminate form so when a function exhibits discontinuity at a or it is undefined at a then we say that these are the indeterminate cases okay so what are the indeterminate cases or indeterminate forms I'd explain this to you in the bridge course so I'll be slightly faster here so indeterminate forms are of seven types yes that is one of the seven types so there are seven indeterminate forms that arises because of your function exhibiting any of these two characteristics so what is the first form which is a zero by zero form but remember not to read it as zero by zero you should actually read it as a red as let me write it as a red as tending to zero by tending to zero normally in your books the word tending to is missing that doesn't mean you'll stop reading it as tending to zero by tending to zero okay so read this in your mind as tending to zero tending to zero form second indeterminate form that probably I will not be able to touch in today's class and not be able to cover up this I'll only be able to cover up few forms under zero by zero is infinity by infinity form okay third one is third one is infinity minus infinity form fourth one is tending to zero okay I'll write it as zero into infinity but read it as zero into infinity but read it as to be read as tending to zero tending to zero into infinity form okay this is how it is read okay fifth is one to the power infinity form now again it is not exactly one to the power infinity because exactly one to the power infinity would be one it is actually tending to one please read this as tending to one to the power of infinity okay this is actually how it is read next is tending to zero to the power tending to zero so please read this as tending to zero to the power tending to zero exactly zero to the power zero is also undefined in maths okay and the last form is infinity to the power tending to zero read as infinity to the power tending to zero okay so as you can see this chapter is not small what primarily you would have covered in school is just the zero by zero form and of course only ncrt level or school level stuff zero by zero itself is a very lengthy concept under limits okay then comes infinity by infinity infinity minus infinity tending to zero into infinity tending to one to the power infinity then tending to zero to the power tending to zero infinity to the pardon it was seven indeterminate forms it actually takes me around three classes to finish it but I would try to do you know cover up as much as I can okay so before we start before we start with zero by zero form I would like you all to understand algebra of limits have you all copied this down any question here anybody this is just a quick recap of whatever we had done in the bridge course now we'll talk about algebra of limits algebra of limits algebra of limits are the rules under which limits work okay so of course when you learn certain operations there is a rule that governs those operations right so they're all happening together what rules you basically apply so let's say you know how to add you know how to subtract you know how to multiply you know how to divide but when all of them come together what is the rule that basically governs the operation that is what we are going to talk about under algebra of limits so the first thing that I'm going to talk about is limit of the sum of two functions f and g okay or difference of two functions is given by limit of f of x plus or minus depending upon the case limit of g of x as x tends to a okay now here I would like you to spend some time because this rules looks to be very simple okay but I have seen in J is some very tricky questions have been formed on this rule or some very tricky questions can be formed on these this rule so please note the following things very very carefully or let me ask this like a question to you okay question is think very carefully okay let me write points to be noted or points to be kept under consideration the first thing that I would like to ask you is let us say let us say if limit of some or difference or whatever is the case let's say I take the sum first difference is also the same thing if this limit exists think carefully and answer okay think carefully and answer does it mean both the limits f and g have to exist or my question can be seen in this way if this limit exists can it happen that can it happen that the limit of f of x limit of f of x as x tends to a does not exist and limit of x tends to a g of x also doesn't exist can it happen in this way some of you are saying no some of you are saying yes first of all I would like to hear from the students who are saying yes can you give me an example where the limit of the function f as x tends to a doesn't exist limit of g of x as x tends to a also doesn't exist but the moment you add the two function f and g its limit starts to exist as x tends to a okay Aryan is saying yes it can happen Akshad is also saying yes it can happen Shraddha is saying yes no both Shraddha you will get run out yes no both calls are different dangerous not sure okay you change your final answer is no Shraddha right okay so now please let me tell you that there can be a case where both the limits doesn't exist but when you add the two function the limit starts existing a classic example and how could you miss this example is when you have your f of x as gif and your g of x is fractional part how could you miss this example this is such a classic example for the same if somebody asks you what is the limit of gif as x tends to 1 what will you say what will you say doesn't exist correct gif you have all seen gif graph you have all seen it's like a step correct no so at 1 left hand limit is 0 right hand limit is 1 so limit doesn't exist can't you cause it 0 to so this isn't exist right fractional part even fraction part diagram if you see it is like this correct this is the graph of fractional part of x so left hand limit of fraction part as x tends to 1 is 1 right hand limit of fraction part as x tends to 1 is 0 so left hand limit right hand limit are different for fraction part so even this doesn't exist even this doesn't exist okay but the moment you add it but the moment you add it f of x plus g of x will become will become x correct and your limit of x as x tends to 1 that is existing and that is 1 okay so a question may arise a question may come in any competitive exam saying that f and g are such functions whose some limit exist exists so what can you comment on the following options or which of the following options do you think is can be true so you may find such an option as one of your correct options okay okay can it happen can it happen this is your situation number a situation number b is can it happen that one of the limits doesn't exist let's say this does not exist and this exists can it happen that one of the limits f or g whatever okay let me write or vice versa or otherwise also vice versa also you can write can it happen that one of them exists other doesn't and your some of the limit of f and g or you can say some of f and g limit as x tends to exist can this happen can this happen okay Prakul Prakul gives his answer very good if yes I would like to hear from you a case where one of the limit doesn't exist other exists and when you add the two functions then the limit also exists can it happen like that okay kinchuk is saying yes so here's the first person so fast to say yes so kinchuk can you give me an example for the same I'm sure you would have come with an example that's why you are saying like this x plus one plus one by x minus one okay so let me write it this is your which function okay so there are two functions x plus one I think this is what you are calling as f of x correct and g of x is is what you are calling as one by x minus one okay and you're talking about extending to one okay very good so limit of extending to one of f of x is to that is existing very good and limit of g of x as x tends to one does not exist okay now what happens when you add them what happens when you add them what does it become it becomes x plus one plus one by x minus one which is x square minus one plus one by x minus one this becomes x square by x minus one do you think the limit of this as x tends to one exists no kinchuk correct so your example doesn't work here and in reality you cannot find such examples because it cannot happen like this right so the conclusion here is so I'm raising a kinchuk example the conclusion here is if your limit of f of x plus g of x exists then it can never happen that limit of one of the function f or g whichever you want exists and the other one doesn't okay so this case is not possible okay so if you see an option like this so they'll say okay limit of f of x plus g of x as x tends to a exists then which of the following are possible and you find the option where they say f is limit of f of x doesn't exist limit of g of x exists or if you see an option like limit of f of x exists limit of g of x doesn't exist please do not mark that option okay please do not mark that option so it is not possible for this to happen okay so remember for your limit of f of x plus g of x to exist either both the limits must exist which is an obvious case that is why I did not write it so I can let me write it and see case so either your both the limits exists okay or in certain cases if they both don't exist then there may be a possibility that the limit will exist now don't get me wrong over here I'm not saying that I'm not saying that if both of them do not exist this limit have to exist I'm not saying that I'm saying that if this limits f of x plus g of x as x tends to exist then they can exist even when both of them are non-existent so there can be cases like this when both of them are non-existent and this limit is existed are you getting my point so don't get me don't get me wrong here thinking that oh sir is trying to say that both must definitely not exist or both will definitely exist for me to have this limit existed no if this is true the C condition then this is always true okay but even if both are non-existent there may be a case there may be a case mark the word maybe where f of x plus g of x limit as x tends to a exists okay is this clear now second important point to be noted which I had discussed with you in the bridge course if you have infinitely many terms added and all of them are tending to zero then this funda will not work if you have infinitely many terms okay the example which I cited in the bridge course I still remember that example I'll I'll take that example let's say I want to evaluate the limit of 1 by n or 1 by n square 2 by n square 3 by n square all the way till let's say n by n square and tending to infinity okay now one student who thinks or who doesn't know the you know the caveat over here what he will do is okay I'll evaluate the limit of each one of them separately so he'll evaluate limit of 1 by n square he will evaluate limit of 2 by n square and he'll continue doing till he goes to limit of n by n square and tending to infinity and what he'll see he'll see oh and is a very large number 1 by n has n square will be 0 this will also be 0 okay this will also be 0 so dot dot dot all of them will be 0 and he will write his answer as 0 and his marks will also be 0 why his marks will be 0 what went wrong in this approach what went wrong in this approach no it's not minus one the answer is actually the answer is actually half okay yes absolutely Gayatri this is not a term which is see if there are finitely many terms like this let's say 100 200 5000 1 lakh 1 million but they're finite then still you can think of writing 0 no problem with that but are the terms limited here no they're infinitely many terms and each one of these terms are tending to 0 these terms are all tending to 0 my dear students these are all tending to 0 so actually you have I think I need to switch off my camera for some time yeah so here you have infinitely many terms which are tending to 0 so it becomes actually tending to 0 into infinity form right the answer for this is anything it could be anything it depends upon how small is your zero and how big is your infinity it's an indeterminate form we cannot write this as zero into something right okay so what is the right way to solve this this is the wrong way to solve this the right way to solve it is if you know your sum of 1 to n terms you use that sum okay I hope you all know your g a p formula from your class 10 okay so this becomes n into n plus 1 by 2 n square cancel off this n divide by this n which is in the denominator both the numerator terms so it becomes n by n plus 1 by n n by n is 1 and 1 by n is 0 so your answer will become half your answer will become half okay so this is the right way to deal with this problem this is the right way to deal with this problem okay good next we'll talk about rule number two if you have product of two functions f and g okay then you can evaluate this limit by using the formula or using the limits of each one of these functions separately and multiplying the result okay now this is subject to the fact that it should not lead to any kind of indeterminacy okay oh sure sure pukul pukul wants me to revisit the previous slide just give me a second pukul system is not very fast way yes anything that you would like me to answer here or is just it is just that you want to copy something done pukul oh thank you thank you yeah now here also some potential questions can be framed so let me ask you important point to be noted if your limit of product of two functions f and g as x tends to a x tends to a exists okay then let me give you some options then choose the correct option then choose the correct options correct options option a limit of f of x as x tends to a now may not exist and limit of g of x as x tends to a also may not exist option b limit of extending to a f of x may not exist and limit of g of x as x tends to a must exist or may exist option c limit of f of x exists and limit of g of x limit of g of x as x tends to a may not exist an option d both of them must exist which of them could be your right options or option think carefully and then answer okay so if b is correct c will be correct that that everybody can make out from common sense right b and c will come together so either both are rejected or both are accepted and then a and d you have to make a call yeah think think guys i want you to think some of you are answering without thinking it's a very simple you know property that you may have read it hundred times while solving while reading through the theory of your ncrt but you will not have you were well analyzed it this is what jay wants you to do you want they want you to analyze okay they're very good so they've madhu mati shadha stuti prakul gaitri kinshuk i got answers from you people what about others i want everybody to participate okay ditu very good rohan okay okay okay okay okay one more minute i'll give you please think if you want to correct your answer do so and then in one minute's time we'll discuss it we'll discuss kinshuk we'll take examples also okay now the verdict is all of the options can be correct surprised right some of you are surprised so all the options may be correct so you may have the product of two functions limit existing even when both of them do not exist or even if one of them doesn't exist another exists and of course when both exist it has to be so these paka karaka i don't need i don't see a reason to explain d but i'll explain a b and c with some example a i can say let's say i have a function like this example for a let me write it like this example for a if you have a function like this which is defined as two when x is let's say greater than equal to zero and minus one when let's say x is less than zero definitely what is the limit of this function as x tends to zero or you can say one only not going to write a minus one and one okay if i ask you what is the limit of this function as x tends to zero what will you say you will say sir it doesn't exist correct because left hand limit is one and right hand limit is two correct yes or no what will you say when i say what will you say when i say what is the limit of this function as x tends to zero you will say sir left hand limit for g as x tends to zero is two and right hand limit of g of x as x tends to zero is one so in this case also the limit of g of x as x tends to zero does not exist correct all agree that both the limits do not exist but see when you multiply it when you multiply these two functions remember only those parts which are under the same domain will be multiplied so greater than equal to zero part will be multiplied together less than zero part will be multiplied together so it will be two when x is greater than equal to zero and again a two when x is greater than zero and in this case you would realize that the limit of the product as x tends to zero very much exists and that is equal to two okay so remember when both of them do not exist just like our previous case then the limit of the product may exist may exist right i'm not saying when every time both of them will be non-existent their product will exist i'm not saying that i'm saying that even if the limit of both the functions individually do not exist limit of the product may exist may is the word here okay so please do not mark this option as incorrect because it is it may be existing it may not be okay for b yes prakul has given a very good example for b for b you can take an example of 1 by x in sin x okay that's a very good example uh prakul so example for b example for b you can take your f of x to be sin x okay we know that limit of sin x as x tends to zero is zero and you can take your g of x to be 1 by x and we know that limit of 1 by x as x tends to zero does not exist okay but when it comes to their product when it comes to their product sin x by x limit as x tends to zero becomes a 1 okay and you can swap their positions to justify c so as i told you if b is correct c will also be correct so c is also correct and d of course when both of them are existent the product limit will also exist for sure okay can it happen that both of them exist and the product limit doesn't exist can that happen can that happen f of x limit exists g of x limit exists but their product limit doesn't exist can that happen in that that way it cannot happen it cannot happen okay so these are the tricky questions that you know j can frame and confuse you people okay let's move on to the property number three property number three this is as good as limit of f of x as x tends to a divided by limit of g of x as x tends to a now again let me tell you it should not be that your denominator limit is becoming zero it should not be that this should be becoming zero okay if it is becoming zero then your numerator limit should also be becoming zero then only this problem can be further solved of course but not by this law by some other law are you getting my point but if your numerator limit is giving you some non-zero value and your denominator is giving you zero then this case is a case of limit does not exist are you getting my point for you to evaluate limit it should be one of the seven indeterminate forms now which of the seven indeterminate forms has a zero in the denominator only one zero by zero form right so if your numerator is non-zero and denominator is becoming zero then your limit cannot be evaluated you cannot find out the limits for such case you can see limit does not exist but if you feel that you have a function f by g whose limit exists then you must have limit of your numerator function also tending to zero as x tends to zero if your denominator limit is becoming zero it cannot happen that top is non-zero and bottom is zero both of them zero then there is a scope of it getting solved but don't think like that it will definitely get solved there's a scope of getting solved right so we can save such a case but if numerator is a non-zero quantity let's say five denominators can becoming zero then that case cannot be no sort are you getting my point okay next is next is if you have any function multiplied with a constant then constant is just multiplied to the limit result of the function okay so you can always bring out the constant out of the picture next if you have a limit of a function which is modded then it is mod of the limit of the function as x tends to it okay please keep noting down these properties they will be very helpful when you are solving a complicated question so if you have limit of f of x raised to the power g of x this is as good as limit of f of x as x tends to a raised to the power limit of g of x as x tends to it now this is subject to the fact that this should not be a zero by zero form sorry this should not be a tending to zero to the power tending to zero form okay and this should not be tending to one in fact you can't evaluate the limit of g in that case so it should not be tending to zero to the power tending to zero form if it is the case like this then such a limit then such a limit can be evaluated by further use of some other method but not by this simple rule okay yeah so tending to zero tending to zero is another indeterminate form it may give you different answers we'll see that when we are doing this form prakul i think check your list of indeterminate forms this is this might be the sixth one or the seventh one okay we'll talk about it how to solve these kind of forms zero to the power zero is not to be considered as one many people think sir anything to the power zero is one right no zero to the power zero is exactly zero to the power exactly zero is not defined in maths are you getting my point tending to zero to power tending to zero can be found out now people ask me say won't it be zero always no see this zero is to be read as now something or you can say a one let's say one divided by a very large quantity infinite quantity okay and this is to be read as a very small quantity right if you're finding in finite a throat of a small quantity that can become a finite answer are you getting my point if you're finding in finite a throat of a small quantity it can become a finite answer are you getting my point okay we'll come to it the seventh one is the property which is meant for composite function yes kinship that is correct that is correct that is something which i'll i'll talk about in your tending to infinity kind of a limit okay so if you want to find out what is the limit of f of x to which g of x is fed as an input this is called this is to be read as f of g of x what will you read it as f of g of x many books will write it as fog okay so if you're evaluating the limit of fog as x tends to a okay this is given by limit of f of sorry it is given by f of limit of g of x as x tends to a provided provided provided let's say i call this number as l provided f of l exists okay so i mean i can just give a dummy example here let's say your f was tan and your limit of g of x as x tends to let's say something gave you pi by 2 then tan pi by 2 is undefined so in that case we will not be able to find the limit by use of this simple law or by the use of this simple algebra then you have to apply you know other complicated panda which we'll learn later on is this fine few special examples under this i can give you is the ones which are regularly seen if you want to evaluate some examples i can give you if you want to evaluate log of a function or ln of a function whatever then it is log of the limit of the function now again let me tell you this answer should not be a negative value else log will not work log works only for positive arguments so it should not happen that you evaluated the limit of f and it came out to be negative five then this limit does not exist okay another example would be a exponential kind of functions so if you have e to the power f of x or a to the power f of x then you can evaluate the limit by raising f of x limit to the power of it okay so these are some examples that you normally come across when you're dealing with questions is this fine is this fine okay eighth one i have you all copied this can i move on to the right side of the board if you have copied it done okay eighth one eighth one it's a simple property it says that if a function f is less than equal to g in the neighborhood of x equal to a in the neighborhood of x equal to a now people ask me sir what do you mean by neighborhood of x equal to a neighborhood of x equal to a means a minus delta a plus delta closing role okay where delta is an infinitismally small quantity okay then even your limit of f of x as x tends to a will be less than equal to limit of g of x as x tends to it okay so it just says that if your value if the function f is always below g of x in the neighborhood of a then even the limit value of f of x will be lesser than limit of g of x value as x tends to okay simple this is not the this is basically you can say half of your sandwich theorem i'll talk about sandwich theorem little later on when i'm deriving your trigonometric limits so please copy this please copy this and we'll take some questions uh you know based on this we'll take some questions based on this yes your inequality will depend on the given functions absolutely yeah dev has a question yes yes everything kinship don't worry a to z i'll cover this topic okay yeah kinship dev has a question if limit of f of x is not defined or doesn't exist then oh you're talking about this question dev this is subject to the fact that they exist okay i'll add them i'll add this point if they exist happy now good point actually i should have written it very good point let's take questions now let's take questions again this is just testing your basics just testing your basics i'm not yet started talking about methods to evaluate limits i'm just testing your basics okay now here all the brackets that you see are greatest integer function brackets okay so i'll put the poll on one sold please press on the poll button very good two of you have responded so far options are given no kinship zero one minus one does not exist okay arian fine noted down your answer no no no i'm getting different different answers some of them have given a some of them be them some of them see and some of them be okay we'll stop this poll in another 30 seconds 11 of you haven't voted yet last 15 seconds five four three two one go okay now end of poll let me share the result with you most of you have gone with option number a okay that is zero let's check let's check so in this case let us evaluate the left hand limit first let us evaluate the left hand limit the limit of this function as x tends to pi by two minus okay now sign of x when you're slightly less than pi by two can i say you're slightly less than one correct sign x when you're slightly less than pi by two let's say you your angle is 89.99999 sign of that will be slightly less than one even if it is 0.999 what will gif do to it it will make it fall to zero correct correct cos slightly less than 90 degree it is a positive small quantity correct i hope you can relate it to the graph of cos x this is pi by two slightly left to pi by two let's say pi by two minus cos will be a very small positive value correct gif of that will also become a zero right so this will also become a zero so you'll end up getting gif of one by three that is anyways as zero so left hand limit is zero for sure what about right hand limit what about right hand limit so limit extending to pi by two plus okay now when you're slightly more than pi by two sign is slightly lesser than one again because it achieves its maximum value of one at pi by two so here and there about 90 degree left or right of 90 degree its value is going to become sulpa less than one so its gif will become zero again okay but what about gif of cos x now the moment you cross pi by two let's say pi by two plus cos is going to become a very small negative number but gif of that small negative number will become a minus one okay so it will become this by three anyways this is gif of two by three that's again zero so left and right hand limit both are equal and both are zero so undoubtedly your answer is option number eight okay now a serious j e aspirant will ask this question what if your gif bracket was outside for example let's say I change the same question I would like you to tell me the answer for the same if I had asked you what is gif of limit extending to pi by two of gif of sin x minus gif of cos x plus one plus one whole divided by three what will your answer be in this case sin pi by two kirtana see this is a graph of sine sine graph right so this is your pi by two so this value is one slightly more than pi by two you'll be at let's say 0.99999 whatever the moment you take gif of it won't it drop to zero yes so what is the answer for this so shambhu says okay rohan okay kinshuk okay very good the answer here is it will not exist because the inside limit itself doesn't exist so why should we talk about the c for any function which is a composition of function to exist you learn this later on in 12th let's say I talk about f o g to exist first of all g must exist okay if inside most function itself is not existing then we don't care about what is happening to the overall function right so I'm talking about a gif case over here where I have two functions one is the gif other is the inside limit okay if the inside answer itself is not you know existing then we don't care about the overall we say it will not exist so answer this it doesn't exist okay now people would say zero for this also and they would get it wrong this is where j will trick you this is where j will trick you shadha just now we saw no if you just ignore the outside gif it will become one by three here and it'll become two by three one by three two by three are different no I changed the question I've brought the outermost gif outside the limit okay so for this part which I'm showing with curly brackets your left hand limit was one by three right hand limit was two by three so limit itself doesn't exist right so okay so please be careful of these small tweaking in the questions it has tricked a lot of people okay let's take this question okay where is that question okay I'll write down the question for you sorry limit of extending to minus seven gif square gif x square plus 15 gif of x plus 56 whole divided by whole divided by sine of x plus seven into sine of x plus eight I will not give you any options I would like you to give me the answer for this question this bracket signifies gif so the square bracket that you can see is gif so again I'll repeat the question limit of extending to minus seven limit extending to minus seven of gif x whole square plus 15 gif x plus 56 upon sine x plus seven into sine x plus eight okay Akshath very good yeah there hence the question actually the question is primarily because it is a zero by zero form yeah so limit exists what is the answer ah no I don't know I have not seen the module that carefully it may be there okay Ruchita very good what is that kinshukh I didn't get you zero by zero and zero by zero and zero by zero is not an answer zero by zero is just the form you can't write an answer of a limit as a zero by zero it just tells you the form what is the answer for that zero by zero is what limit helps you to find out limit is the tool which helps you to get answers to zero by zero form okay so three of you have responded so far one is Ruchita other is Akshath and other is Prakul okay Gayatri they very good okay kinshukh okay let's discuss this in the interest of time okay uh we can all see that your numerator is factorizable isn't it correct me if I'm wrong your numerator is factorizable denominator let it be as it is okay now let us evaluate what is the limit of this function as x sends to minus seven minus minus seven minus means you are slightly less than minus seven probably minus seven point zero zero zero zero one you're slightly less than minus seven less than minus seven okay slightly less than minus seven now if you're slightly less than minus seven what will be the value of gif of x you say sir simple it will be minus eight okay what will be the value here that will be minus 8 plus 0 correct what will happen to the denominator denominator this term would be tending to zero term I'll just write it like this I hope you can understand what I'm trying to write and this will be tending to tending to sign one now here the answer is actually zero it is not a zero by zero it is not a tending to zero by tending to zero form guys and girls pay attention this is actually exactly zero by tending to zero okay and this is not an indeterminate form its answer is plain and simple zero and nothing else because your numerator is exactly zero so it doesn't matter whether your denominator is a small positive number a small negative number or a number tending to zero the whole answer is exactly zero it is not an indeterminate form please do not get this please do not confuse this with a tending to zero by tending to zero form your numerator is not tending to zero my dear it is exactly zero getting my point if it is exactly zero then don't try to apply anything any formula and all plain simple zero that's the end of the game right so I wanted to see whether that concept of tending to zero by tending to zero are you able to distinguish it from exactly zero by tending to zero of course exactly zero by exactly zero cannot be evaluated correct so tending to zero by tending to zero is an indeterminate form which can be evaluated exactly zero by tending to zero is pukka zero force no no no second thought about it okay so left hand limit is zero what about right hand limit right hand limit if you see as x tends to minus seven plus minus seven plus seven plus minus seven okay minus seven plus means you are slightly higher than minus seven really minus six point nine nine nine nine kind of a thing right so if you take a gif now come back to this step if you take the gif of minus six point nine nine it'll give you a minus seven so it'll becomes minus seven plus seven by minus seven plus eight by a quantity again tending to zero by quantity again tending to one I believe sine one I believe but it is a case where your numerator is exactly zero denominator may be tending to zero so this is a case of this is not a case of indeterminate form this is a case of just writing zero as the output are you getting my point so is the left hand limit right hand limit the same you'll say yes sir very much they are the same so your answer for this question is zero any questions here anybody okay so now you would be realizing that there is a lot of fine tuning happening with your understanding yes yes magna you need not factorize it I factorize it just to make it very clear evident but without factorizing also if you do you'll end up getting zero on the numerator anyhow whether you are going to minus infinite minus seven minus or minus seven plus yes of course prakul but provided your limit of the function should not be tending to zero okay now we'll be talking about methods to evaluate limits methods to evaluate to evaluate limit of a function so if you realize your function is a continuous function at extending to a or x equal to a my bad if you realize your function f of x is continuous at x equal to a continuous at x equal to a means no breakage no whole at x equal to a then you can evaluate your limit by just using the formula f of a okay now how would you come to know a function is not continuous or a function is continuous at x equal to a of course one is by making the graph but that is not a easier route is this a difficult functions will come how do you expect you how do you expect us to make graph and all for that okay I understand but at least you can try to put the value of a into the function and see if it is giving you a you know zero by zero or infinity by infinity if all those indeterminate forms do not come up that means it's a case where your function was continuous at x equal to a correct for example let's say if I give you this limit to evaluate x square x square plus two x plus three and I say evaluate the limit of this as x tends to one the moment you put one in the place of x you realize that you end up getting a six six is a you know finite quantity so this answer will also become the limit of the function okay so if you end up getting something which is you know you know weird like zero by zero or infinity by infinity or even tending to one to the power infinity or any of the seven indeterminate forms there you should realize that this is not a case where the function is continuous at x equal to a so I cannot evaluate it by substitution okay so as I already told you in the bridge course substitution may not be the first step of any problem but it definitely is the last step of every problem okay so your teacher might not give you any question where you have to apply substitution from the word go right it is like you know giving three b's to you so they'll not they'll not give you marks like that but yes having said that the last step of every problem solving requires substitution what is the exact purpose of limit that is a very good question exact purpose of limit is when you are evaluating dy by dx that's your derivative dy by dx is what dy is a quantity which is tending to zero dx is also a quantity which is tending to zero so when you are evaluating dy by dx means your entire derivative is actually limits nothing else getting the point so limit is to see how is your numerator changing with respect to the denominator when these changes may be either very large or very small that is the purpose of limits very good question quick kinship okay I think you are not there in the bridge course when I talked about it are you there or you are not there okay okay next is your method of factorization this is called method of substitution let me write it this is called method of substitution method of substitution next is your method of factorization method of factorization method of factorization method of factorization is mostly used when you have two functions f and g where you realize that when you put a into f you get zero and you put a into g also you get zero implying that x minus a is a factor of factor of f of x and g of x okay one more thing I would like to highlight over here even though I'm using the word factor it is not necessary that when you put a in f and you get zero and you put g in a and you get a zero that means x minus one has to come out as a factor no I'm not I don't mean to say that I don't mean to say that for example let's say your f of x was e to the power x minus one correct and your your denominator was let's say tan x and a was zero a was zero so putting zero in pace of x will give you a zero putting zero in pace of x in this one will also give you zero that doesn't mean x is a factor of e to the power x minus one or x is a factor of tan x I don't mean to say that if at all you can take out a factor of x minus a from f and g function okay in other words so people say why don't you say f and g are polynomials then it will make more sense no even for non polynomials also some factors can be pulled out so I don't want to make both the statements are you getting my point so I'm basically choosing my words very carefully I don't want to call f and g polynomials because it is not limited to polynomials x minus a could be pulled out as factors even when the expression is not a polynomial for an irrational function probably irrational functions are not polynomial okay so what I'm trying to say is if your f of a zero and g of a zero and x minus a can be pulled out as a factor from both f of x and g of x then you can write this particular expression like this okay that means x minus a could be taken out as a factor leaving behind capital f of x you can call that as your quotient if you want to and leaving g of x capital g of x as your quotient for the denominator term and then you can cancel off this factors and then put your a in place of x over here and here provided you again do not end up provided you again do not end up in an indeterminate form so this should be a finite answer okay are you getting my point so if you get a zero by zero again that means again there is a scope of x minus a to be pulled out as a factor or probably you can have other ways implemented or you know apply to it to get the job done are you getting my point not a new thing for us we have done few examples on this in our bitch course also so we'll take some questions directly on this we'll take some questions directly on this okay let's take this question x tending to root 2 x to the power 4 minus 4 upon x square plus 3x root 2 minus 8 I'll put the poll on done let's finish this up in one minute you all have done enough of factorization based questions with me in the bitch course both are polynomials thankfully so method of factorization can really help us in this in this problem one person has answered so far our dear students last 30 seconds for this please vote simple question to solve last 15 seconds five four three two one go 16 of you haven't voted yet okay anyways most of you have gone for c most of you have gone for c let us see whether c is the right option or not numerator is definitely factorizable with a factor of x minus root 2 so you can factorize this as x square plus 2 into x square minus 2 x square minus 2 can be further be factorized like this I'm not writing every step over here our denominator if you see this term can be split up as 4x root 2 minus x root 2 minus 8 so if you take an x common it'll be x plus 4 root 2 and if you take a minus root 2 common here it'll become x plus 4 root 2 so it'll become x minus root 2 times x plus 4 root 2 x minus root 2 x minus root 2 that is the problem creating factor I always call this as a problem creating factor because this is the one which is responsible for creating a zero on the numerator and a zero in the denominator okay you can check it is actually a zero by zero form to begin with now again I'm writing zero by zero but how would I read it tending to zero by tending to zero it is not exactly zero by exactly zero exactly zero by exactly zero nothing can be done about it exactly zero by tending to zero that is zero no need to do any kind of a method for that but this is tending to zero by tending so there we need to apply our methodologies so once you have got it you can just go for the substitution so when you substitute you end up getting 2 plus 2 this will become root 2 plus root 2 and this will become root 2 plus 4 root 2 which clearly gives you 4 times 2 root 2 upon 5 root 2 so root 2 root 2 gone answer is 8 upon 5 okay next okay try this one now prima facie it is actually infinity by infinity prima facie means from the first observation it looks like infinity by infinity yes but let me tell you later on you will realize that these indeterminate forms are inter convertible okay so infinity minus infinity can be changed to zero by zero also okay try this out I'm going to wait for you to answer let me give you options sorry let me give you the poll no no no normal brackets kirtana they are normal brackets unless until stated unless until stated don't treat square brackets to be gif it will be mentioned in the question that any square bracket is gif then only use it as gif else they are plain and simple normal brackets only one person has answered so far okay very good let's finish this up in another 30 seconds five people have responded so far oh okay last 10 seconds five four three two one go come on guys 17 of you haven't voted yet pass pass pass pass pass pass okay let's let's discuss it by the way most of you have gone for d does not exist okay let's check let's check let us check so here first of all I'll take the lcm by the way even before doing that I would like to factorize this and this is factorizable as x minus one x minus two okay I would like to take the lcm here first lcm would be x x minus one x minus two square correct so this will give me x minus one minus this will give me x x minus one oh sorry x x minus two okay let us see what does the numerator simplify to so this will give you negative x square plus two x plus x which is three x and you will have minus one negative x square plus two x which is three x minus one is this fine any questions any questions in these steps so far anything that I have missed please highlight no no no problem okay now the numerator if you see as x tends to two is minus four plus six minus one which is not a zero term and denominator is becoming zero so you have a case where numerator is not zero and denominator is zero so it don't start applying your method of you know factorization on this it's a case where limit does not exist plain and simple so please unnecessarily do not apply limits to such cases which were not indeterminate forms at all it was a case of limit doesn't exist simple as that okay if your numerator was coming zero and denominator was coming zero then there would have been some scope some scope would have been left I'm not saying even then your limit must exist no but this is a case which is scopeless or hopeless we cannot do go any you know further from that because numerator is the non zero quantity denominator is zero quantity limit doesn't exist option number t is correct danta was correct next I did this problem yeah let's do this question limit of four x cube minus x square plus two x minus five upon x to the past six plus five x cube minus two x minus four extending to one I'll put the poll on three minutes is what I'm going to give you for this okay one person has responded two people have responded very good nice nice nice nice I don't know the answer aren't even I'm also solving along with you we'll see we'll see which option is correct okay last one minute last one minute for this see first thing that you would all notice is that it's a zero by zero form right the moment you put x equal to one it cannot be solved by substitution that's for sure everybody would have realized it so when you put a one you get four minus one plus two minus five upon one plus five minus two minus four that's six minus six zero again six minus six zero it's a zero by zero form okay now let me tell you it is zero by zero exactly when you're putting x equal to one right so many people ask me sir you only told exactly zero by exactly zero we cannot do anything right but x is not equal to one x is tending to one see here x is tending to one so instead of being exactly zero by exactly zero it is actually tending to zero by tending to zero so we can do something about it actually so definitely both are polynomials and when you're putting one in both of them you're getting zero which means x minus one is a factor for both the polynomials so what I'll do is I'll try to bring out the quotient when you divide the numerator and denominator respectively by x minus one so many people would be doing this step so you would divide your numerator by x minus one okay and you'll also be dividing your denominator by x minus one so you have to do this operation okay and yes okay by the way I'll stop the poll here most of you have replied with option D this is the reply of most of it okay let's see whether D is correct and you'll also divide x to the power six plus five x q minus two x minus four by x minus one right this is what you'll be doing and most of you would be using long division method while long division method is good for your school exams but it is very time consuming when you are applying it in comparative exams so for comparative exams we normally mention a method which we call as the horner's method also called as the synthetic division method so I told this method in the bridge course some of you still remember it I can see that so those who are not there in the bridge course for them it's a new thing well you would have learned this in class 10th Tushar sir would have definitely taught horner's method okay so in this method what do we do is let's say I want to do this operation okay so first of all I will write down this polynomial from increasing to decreasing power without missing out any fact without missing out any power so as you can see there is no power being missed out so I've written it like this okay then I will take its coefficient for minus one to minus five so all the coefficients I took down okay now since you're dividing by x minus one write a one over here okay so basically I'm telling you a shortcut for long division okay so horner method or synthetic division is a shortcut by which you can divide any polynomial of any degree by a linear polynomial yes this is a limitation actually this method is not a complete replacement for long division long division method can be used even when you're dividing it by a quadratic or by a cubic or by a bi quadratic okay but horner method is used only when you're dividing by a linear term like this so that is the limitation of horner's method okay now how does it work see the modus operandi for this how does it work first you put a zero over here by default every time every time you put a zero below the very first coefficient that you have written okay add these two what is four plus zero four multiply four with this number that is sitting over here so multiply these two and write the result over here so one into four four write it over there so whatever is the number there let's say if you were dividing by x minus two then two would have come there so two into four eight would have come over here right okay so whatever is that number multiplied with this sum and write over it then again add then again multiply then again add sorry again multiply and then again add that will give you a zero okay this last number that you see is actually the remainder we all know that x minus one is a factor of this four x cube minus x square plus two eight so remainder must come out to be zero that's why zero is coming out and what about these numbers that you have written four three and five basically write them down in a reverse manner with increasing power of x starting from zero that means this becomes your quotient this is your quotient okay see how less you have to write within two lines your long division method is that correct let us apply those who have not learned this method let us apply it again on the second one this one again x to the power six five now remember you have to write down all the terms without missing out even a single power of x so x to the power six is there but x to the power five is not there so what will you do you let zero x to the power five okay then you'll write zero x to the power four then you will write five x cube then you will write zero x square because there's no x square term then you'll write a minus two x then you'll write a minus four okay now bring down these coefficients so one zero zero five zero minus two minus four make this weird symbol over here there's no meaning for the symbol many people say why do you draw this symbol you can draw any symbol you want it is just to separate these terms from this number that's it by default write a zero one plus zero is one one into one is one zero plus one is one one into one is one zero plus one is one one into one is one five plus one is six one into six is six zero plus six is six this into this will be six then this will be four then this will be four and the last will be zero if last one is not coming out to be zero that means you have done some mistake mistake you have done some mistake okay now start writing these numbers from the last one which is the 4 starting from power x equal to 0 so 4 x to the power 0 6x 6x square x cube x4 x to the power 5 so this will become your quotient this will become your quotient see how efficient it is okay I didn't get since the denominators are same for all the options we need not factorize it right oh yes I think you figured out from the options that denominators are all 1990 1990 that's a very good move you can save that you can take that approach and save a lot of time but right now my duty is to tell you the full procedure of solving the question okay so this question as x tends to 1 you can write the numerator as x minus 1 4x square plus 3x plus 5 denominator as denominator as x minus 1 x to the power 5 x to the power 4 x to the power 3 6x square 6x plus 4 this is a problem creating factor gone okay so numerator will give you 4 plus 3 plus 5 which is 12 denominator will give you 1 plus 1 plus 1 plus 6 plus 6 plus 4 okay that is 12 by 19 is your answer which is given by option number option number D so it's a good learning for people who were not there with us in the bridge course this is how you can do quick division okay is this fine let's take up the next question evaluate this limit limit extending to pi by 4 1 minus cot qx by 2 minus cot x minus cot qx now many of you would be thinking are you sure I started trigonometric limits also here actually no actually no yeah I'll not give out the options you will have to type it out in the chat box 1 minus cot cube divided by 2 minus cot x minus cot qx extending to pi by 4 extremely happy to see nearly full turnout even on a test day that's the spirit hats off to all of you understand UT days are very critical but almost everybody is present for the class that's a good sign done okay at shit good okay last 30 seconds okay Shambo very good good Rohan is that it be wrong I don't care about right or wrong I care about you giving me the answer of course don't write anything that you feel like giving me the answer is like synonymous with you making an attempt to solve it that is more important to me okay sooty richita very good no solution is this an equation that you are trying to solve that you're saying no solution doesn't exist that is what you want to say okay there is a solution the limit does exist don't worry breakfast a limit exists okay so let us solve this question let us solve this question see treat it as if you are dealing with cortex replaced with some variable T this is called intermediate substitution so you are making an intermediate substitution in order to convert it to a form which is easy for you to work on correct so what I'm going to do is I'm going to call my cot of x as a T so as X tends to pi by 4 T will tend to one okay so this limit is as good as evaluating the limit of 1 minus T cube by 2 minus T minus T cube T tend to one both will give me the same answer okay now this is much easier to deal with because I can easily see that it is factorizable I'm sure we can factorize the numerator as 1 minus T 1 plus T plus T square what about the denominator denominator let us do one thing let us apply our let us apply our Horner's method okay you're dividing by T minus 1 right you're dividing this by T minus 1 so Horner method what do we do we write 1 okay and we write the coefficients of these power different powers of T minus 1 0 minus 1 2 0 it's very fast actually minus 1 minus 1 minus 1 minus 1 minus 2 minus 2 0 so that will give you T minus 1 minus T square minus T minus 2 so you can do one thing you can cancel out this 1 minus T in this and take away the negative sign from it okay then put your T value as 1 it becomes 3 upon 4 so the answer is 3 by 4 I think only one person got no they ever got it right they ever got it right Rohan got it right Akshath got it right very good well done others made some sign mistakes okay next is method of rationalization