Transcription
Now whenever the child is told, "Son, apply this one formula, all the questions will be solved," the child becomes happy. So my teaching method says, "Brother, give one formula, all questions." And it's not that these questions don't come in the exam, they are full of them, brother, they are full of them in the exam, these questions. Okay, wait a moment. So an uncle was doing shoulder exercises, and he said "Yal," and "Yal" came out of his mouth. Since then, I remembered that brother, "Yal," "Tahalka," "Omelette," all those questions, approximately 10 to 12 questions, those 10-12 questions are superbly important and a bit of a heavy part. So those questions, now watch, after an hour, those questions are starting, son. So those who missed the beginning will have difficulty. Hello dear children, greetings to everyone. Welcome to Science and Fun, where teachers teach with both heart and mind. So here is the Super One Shot of Continuity and Differentiability. There was a lot of demand, and in every video, there was only one comment: "Sir, do continuity," "Sir, do continuous differentiability." So, son, we have brought it for you. You will get complete NCERT coverage, you will get PYQs, just like in the previous lecture, you will get every single PYQ, and the most expected questions from Exemplar, or questions that seem likely to come in the exam, three such questions are very explosive. Okay, brother. This is a very big chapter. Okay? And continuity and differentiability are topics related to the past, meaning Limits and Derivatives. In 11th standard, you all studied Limits and Derivatives, so this chapter is related to that. If I talk precisely, you will see here that continuity is written large and differentiability is written small. So there is a reason behind this. The reason is that in today's lecture, we are only considering continuity. But we will go so deep into continuity that you will not have any difficulty in any question. By the way, let me tell you, this is my claim. Do all these questions, and you will not need to do any other questions. Take this in writing. Then, we will study differentiability in a similar way. So, a proper one shot becomes very difficult. So today I will give you continuity, otherwise, it would have been even later. So today you study continuity, and after three to four days, you will get a power-packed one shot of differentiability. Got it? Okay, so now we will talk about continuity. What is continuity? Let's talk about continuity. See, continuity, or a word called continuous function. Continuous function. Now you all must have read this word, function, in the last lecture, Relation and Function. Let me tell you, I have already taught four chapters properly through one shots. Matrices came in two parts. After that, your Determinant was a proper one shot. Relation and Function was a proper one shot. Then the last topic I uploaded was Inverse Trigonometric Functions. So, we have been studying the word function for the last two one shots. So here too, you now have to study about continuous functions. The same functions that you started studying from 11th standard. The same ones. See, linear, constant, polynomial, logarithmic functions, exponential functions, trigonometric functions, and greatest integer, modulus. We will talk about all of them. So, a function is continuous, meaning continuous. What does continuous mean? When is it called continuous? When is it called continuous? Now, in my layman's language, when a function is set to be continuous. Properly. A function, a function is said to be, is said to be continuous if. Now, in layman's language, if the pen doesn't lift while drawing the graph. Now, children misunderstand this. Sir, if we don't even know how to draw the graph, meaning we can't even lift the pen to draw the graph, then the function becomes continuous, right? No, brother. When you are drawing its graph, if the graph is being drawn in one flow, without lifting the pen, then that function will be called a continuous function. When a movie starts, a disclaimer comes. So, taking one second, one second less, taking 10 seconds, I will talk about a disclaimer. We are at a stage where I can teach you this chapter in two ways. First, apply the formula, get the answer. Those children are happy who just want marks. Second, first build the concept, derive the formula, explain the concept, and then tell the formula, and then solve the questions. You will get to meet these paths later. This is a shortcut. This is a bit of a long cut. But this long cut, this slightly hard work method, will be very helpful later when you sit for exams like CUET. You have to crack MCQs. And from this chapter, from this topic, so many MCQs are not asked in 12th standard. But if asked, you will tick it at a glance. In this method, you will have to apply the formula. In this method, for small questions, you will tick, tick, tick. It will be like this. So, I know some children will say, "Sir, this one." But the majority of children will say, "This one." I was telling you that I will adopt this path because it's in my blood. Whether I teach offline, whether I teach live, whether I make paid batches, or whether I make recorded sessions, I cannot go directly. It's not in my blood. Okay? So, whoever feels that they need the formula from here, then brother, just skip directly to the formula. I'm telling you beforehand. I've given the disclaimer. And whoever wants the complete path to tick at a glance, later on, or even in this year's 12th board exam, then you will have to listen with a little patience, and then we will start the questions. And the questions are such that they are "chummeshwari" in my language. Okay? So, a function is said to be continuous if its pen doesn't lift while drawing the graph. So, do we have to draw the graph of all functions? Wait and watch. Wait and watch. Let's come. Now, for example, let's talk about polynomial functions. Polynomial functions include constant functions, meaning those with degree zero. Now, you can see clearly, when I draw its graph, when I draw its graph, the pen doesn't lift. Check it. The pen doesn't lift. See, this is a constant function, meaning fx = 5. Okay, brother. fx = 5. So, this line will pass through 5. Suppose this is -5, then it will go from here. Suppose this is 2, then the line will go from here. Whenever you draw its graph, the pen doesn't lift. So, the constant function is a continuous function. Then comes the linear function. Sir, linear with explanation. Linear function. Let's assume it's of two variables. Suppose, suppose you studied all this in childhood, that y = x + 2. Let's assume this. Haven't you studied this? You must have drawn graphs of linear equations in two variables. See, you must have drawn graphs like this. You must have put values. If you put x = 0, y = 2. You do it like this. Suppose if you put 1, then it's 3. If you put 2, then it's 4. It's like this, right? So, when we talk like this, x = 0, then y = 2. At 1, it's 3. At 2, it's 4. So, it will form a straight line. It will never happen that the pen lifts in the middle. It's not like you go here, it becomes like this, and then you lift the pen and draw it. No, right? What will happen, son? It will be a straight line, drawn without lifting the pen. So, remember that a linear function is always a continuous function. Okay. Then comes the quadratic function. Quadratic. fx = x^2. Quadratic function. It has a parabolic shape. Parabolic shape. The graph of x^2 goes through the origin. If it's x^2 + something else, then you can see clearly. This graph, this graph can be drawn without lifting the pen. So, a quadratic function is also our continuous function. Okay, brother. Then comes the cubic function. You can check. When we draw its graph, this function can also be drawn without lifting the pen. Similarly, a biquadratic function, whose degree is 4, can also be drawn without lifting the pen. Like a quartic function, whose degree is 5. Biquadratic, quintic. Okay, brother. Now, see, you don't know. All polynomial functions. Power 5, 6, 7, 8, 9, 10, anything comes. In all of them, see, this is the same as the W from the movie "Dhamal." So, here, see, I have drawn all these graphs without lifting the pen. So, you can say, now you can say with certainty here that, sir, this polynomial function will always be continuous. Whenever you get a question like, "Tell me directly, is x^2, x^5, x^6 a continuous function?" Yes, it is. It's directly continuous at any value of x. Okay. Let's come, let's come. We will come again. Then comes, children, our trigonometric functions. Trigonometric functions. You all studied in 11th standard. Like, it's a bit blurry because I made it intentionally light so that I can darken it. Whose function is this? Comment quickly. Comment quickly. Whose function is this? It's written here too: sin x - cos x. sin x - cos x. Oh, that's sine and cosine. So, write here. Write in the notes: Only, only sin x - cos x. I'm talking about it. Let's change the color. Suppose I'm talking about the interval from 0 to pi/2. So, between 0 and pi/2, the cos x function is. See, it's continuous. In the interval 0 to pi/2, the function is continuous. Understood? But it's not continuous over the entire real number. It's not continuous throughout its domain. Okay, brother. Look further. Then comes your favorite, favorite function, the log function. You have never drawn the graph of a log function, but it always looks like this. Log function. It's like this. Okay, brother. It's like this. Its shape is like this. So, it's drawn without lifting the pen. So, the log function is also what kind of function, son? A continuous function. And this is the inverse of log. This is our exponential function. See, it's drawn without lifting the pen. So, this function is also, dear children, a continuous function. After this, we will talk about the modulus function. It's a V-shaped function. It's always drawn without lifting the pen. So, it is always continuous. Understand? Now, if you remember all these things, if you have remembered them, if you have felt them, which function is this? Greatest integer function. Which function is this? Greatest integer function. What is this? We will talk about it in the questions too. Anyway, there are not many questions for it in continuity. Okay? If there are, then, for example, here, see, first you have to draw this, then this, then this. The pen will lift again and again. It's also called a step function. The pen will lift again and again. So, such functions are, dear children, discontinuous. Is the point clear? So, the moral of the story, the moral of the story, the moral of the story. Now, let's look here. All our polynomials, all our polynomial functions. If you look at your NCERT, dear children, the first four-five questions are very strange, and the last three-four questions are also very strange. Strange means they are asking you to prove, to tell. Is x + 5 a continuous function? Yes, it is. Is x^2 + 5 a continuous function? Yes, it is. Is 3x - 6x^2 + 5 a continuous function? Yes, it is. Is |x + 1| a continuous function? Yes, it is. Because modulus is continuous. So, we have to tell this directly. It will be a one-marker question. We will tick it. Okay, brother. Now we will talk about PYQs. Has such a question come or not? Polynomial is done. After that, let's talk. Now you all studied trigonometric functions. In trigonometric functions, only two functions, i.e., sin x - cos x, and some functions are not continuous, as we saw, tan x, etc. Now, for example, you studied a signum function. It is also not continuous. x upon |x| is also not continuous. Okay? But, but, but, but, but, wait, wait. This will also depend on the limit. We were talking about all these functions. x belongs to the real numbers. So, throughout the real numbers, these functions remain continuous. But now a question arises: Sir, we saw it from the graph here. Sir, we saw it from the graph here too. Sir, we saw it from the graph here too. Sir, we saw the graph here too. So, will continuity be determined only by the graph? There are so many functions whose graphs I don't even know how to draw, and if I do know, it's very difficult to draw them. So, do I have to remember the graphs of all of them? The answer is no. Come, to explain this, I will ask you a question. Suppose there is a function f(x). It is 1 when x is greater than or equal to 0, and 2 when x is less than 0. I have written this function for you. Now, if I have to check whether this function is continuous or not, how will it be checked, son? If I have to check whether this function is continuous or not, how will it be checked? Come, I'll tell you. See, first of all, we draw a graph and try to derive the formula from it, what is the specialty of continuous functions that they become continuous? Okay? So, let's derive the formula. See, first of all, I'll draw a graph for you. Okay, brother. Now, you won't even need to remember the formula yourself. You will be able to figure out yourself whether it's continuous or not. The formula will be derived. Now, see, x >= 0, meaning it comes here, so the answer is 1. That is, at 0, the answer is 1, and if you put a value greater than 0, the answer is still 1. And if x is less than 0, this is x, right? Less than 0, then the answer is 2. The answer is 2. Not 2 at 0. The answer is 2. Okay, now you tell me. Did I draw the graph in front of you? I drew it in front of you. Type quickly. Did this function become continuous? Did this function become continuous? Check. See, when I drew its graph in front of you, at x=0, it's 1. If you put a value greater than 0, it's still 1. If you put a value less than 0, it's 2. So, you had to lift the pen, right, son? So, this function is not continuous. When would this function be continuous? When would this function be continuous? When, when, when, when, if the answer for greater than 0 also came as 1, at 0 it was 0, and the answer for less than 0 also came as 2, then we would draw a graph like this and move on. Am I saying this right or wrong? So, if this part and that part were the same, then this function would become continuous. Now, see, this is the critical point. This is our critical point. Here it is, zero. If you go to the positive side from zero, this is called RHL, Right Hand Limit. What is Right Hand Limit? Gradually approaching the formula. And if you go a little behind zero, like here, less than zero, greater than zero. You are checking continuity at zero. So, you have to go a little before zero and a little after zero. If you increase zero a little, that is called RHL. If you decrease it a little, that is called LHL, Left Hand Limit. Understood? Now, it says that if LHL and RHL become equal, and along with that, at zero, i.e., f(a) also becomes equal, then the function becomes continuous. To see this, take another example. Like, I have modified this question a bit. Like, 1 when x is not equal to 0, and 2 when x is equal to 0. The function says, if x is not equal to 0, then 1. Not equal to 0 means greater than 0 or less than 0. And if x is 0, then the answer will be 2. Correct? Okay, now, if I draw the graph here. Now, if I draw the graph here. Okay, brother. Quickly. So, see carefully. This is 0. This is 1. And this is 2. Now, see, x is less than 0, i.e., LHL. x is greater than 0, i.e., RHL. So, what do we have to do? Draw like this. Draw like this. And draw like this. Now, if x is 0, then the value of y, i.e., f(x), is not 1. At x=0, f(x) is 2. When drawing the graph, the pen had to lift. Now, understand carefully. Here, if I see, my LHL has become equal to RHL. But to draw this, I had to lift the pen. Only to draw this. If this also came on top of this, then my function would be 100% continuous. Meaning, if x was equal to 0, the answer would also be 1. Then what would I do? I would do this. I would do this. And I would do this. There would be no need to draw this. Understood? But it didn't happen like this. So, the moral of the story is. The moral of the story. The moral of the story. The formula, brother, has been created by itself. It says that if LHL is equal to RHL, and and is equal to f(a), then the function will be continuous. Where 'a' is that critical point. 'a' will be that critical point on which you are checking. Like, in both these cases, our 'a' was 0, because in both cases we were checking at 0. Sometimes we will check at 1, sometimes at 2, sometimes at 3, at -3. So, that is the critical point of the function. Okay. So, if LHL, RHL, and f(a) are equal, then the function can be continuous. Okay, brother. Let's go. Now, let me elaborate a bit more on its formulas. What is the method to find LHL? Limit x tends to a-. Now, LHL means, as I told you, the point 'a', whatever it is. Now, here 'a' was 0. Let's assume in some question 'a' was 1. Then, the thing before 1 is LHL. The thing after 1 is RHL. Got it? So, here, behind 'a', RHL. What will be LHL? And if we want to find RHL, then after 'a', so 'a' negative, i.e., before 'a'. 'a' positive, i.e., after 'a'. And 'a' is equal to 'a'. If all three are equal, then this function will be continuous. Now, sir, how to find it? Son, to find it, we have to go back a bit to 11th standard. Obviously. But I am here, right? Explaining. See, its formula. And if we extend it further, we have to take such a quantity, h, which is a very small quantity, very small, close to zero. You know what this is called? h tends to zero. h is.
Not equal to zero, but A is very close to zero. So we considered such a quantity A, very, very close to zero. And our function, our function, what do I have to put in that function? What to put in the function? A minus H is the formula for LHL. In the function, A minus H has to be put. RAL is the formula. Now when I do two questions, there will be no need for the third, then you will understand yourself what this formula is saying. Okay sir, this came A plus H. In LHL, A minus H, meaning smaller than A. Look, if we assume A is five, for explanation, the critical point is five. So what does LHL say? Smaller than five. So I took H as a very small quantity which is smaller than five. That's why I did 5 - H. And RAL means if you want greater than five, then add H to five. You will move a little forward, a little means very close. And the third came, brother, A. If these three become equal to each other, children, if all three become equal to each other, children, only then will the function be called continuous, otherwise it will not be called. If two out of three are equal and one is not equal, then the function is not continuous. Okay sir, for now, remember all this. Here, if you want to write one thing, it's also good: f(x) f(x), because it will be a function, right? f. These three will be equal, then the function will become continuous. Is the point understood? Is the point understood? Is it clear?
We will come back to this slide. Some points to remember, points to remember. As cases come before us, we will write some points to remember here. Okay sir, now for example, I don't know if the next slide is empty or not. What we do is, we select this a bit and bring it here and start writing from here. Because there are quite a few, slowly, slowly, this will become the most important slide. Brother, remember this very well. Slowly, this slide will become very important. Okay son, look carefully. So first of all, you have seen the formula. First of all, you have seen the formula, and then now we will see points to remember. Now, whether a function is continuous or not, there is no need for a graph, there is a need for the formula. Is this much clear? Give me a thumbs up. Is it done till here? Give me a thumbs up.
So, by spending these 24-25 minutes, I went a little out of the way and made you memorize some functions orally. And tried to derive a formula. Many children must have understood it. If not, then rewind and watch it. And if you don't understand, just questions are starting, PYQs are coming. Now it's just questions, just questions. Now I will make a sensational omelet. Look, come on, it's my favorite topic, man. I enjoy a lot in studying and teaching. Look.
So first let's see the first question. Find all points of discontinuity of F, F meaning function, where F is defined by this. A question scares a lot, all points. So there will be infinite points. Yes, there will be infinite. Wait, but the way to write is very simple. Check at two. Where do you only have to check? At the critical point given to you. You have to check only at two. Correct? Come on, now look, first of all, if I tell you one thing, now I have some experience. Will you take it or not, or should I just do the questions and leave? Brother, you will take some experience from me, right? Now I have some experience that will save your time in the exam, so try to understand that. It might be that two questions will seem a bit strange, but have a little patience, have faith in me. So after the third or fourth question, you will say, "Sir, what a one-shot you have made!" Sir, you will write common yourself. Meaning your fingers, your fingers will become such that, "Man, I appreciate it." That concept will strike your brain, just once, here too and there too, it will strike everywhere. Just watch a little what I am doing and how I am doing it. Look, we have to find three things: first LHL, RAL, and f(x). Among these three, what is the easiest to find? This, this, or this? Your answer will be, "Sir, this." So first of all, one should start from here. So first of all, I will find A. Now A means here A, here it is 2, so I will find f(2). Now where do you find f(2)? Where there is an equals to sign. Where there will be an equals to sign, put X in front of it. Okay sir, now if you put two here, two and three, then what did this function become, brother? Four and three, seven. Correct? Now many children will say, "Sir, just now you talked about solving and ticking." Beta, that would be if, suppose, a question came, "Tell me, is 2x + 3 continuous at x = 2?" Then I wouldn't, sorry, I wouldn't solve anything, I would directly say this is continuous because it is a polynomial linear function. But the function got divided into two things, it got divided into two things, then I will have to make a formula. Okay sir, this is a one-marker question. This is not a one-marker question. Okay, is the point understood? Come on, so what did f(2) come out to be now? 7. Now I have to find either LHL or RAL. Now listen to me carefully, very carefully. Here is equals to. I have already found this. Now, first find the one opposite to equals to. That can be LHL, that can also be RAL. Meaning this is already found, right? Now first find this. Now here the bird's beak is open, son. Here the bird's beak is open towards x. So if the bird's beak is open towards x, then obviously it will be RAL. And if the bird's beak is closed towards x, then children, that will be LHL. Remember this too, some children don't understand. Okay, bird's beak means x is greater, meaning positive, meaning it is greater than that A. It is greater than that A, so it became RAL. X is smaller than A, so smaller means we are going to this side, going to the left side, so it became Left Hand Limit. Okay sir, so now you have found this one, the equals to one. Now find the opposite one. By chance, here the opposite one is RAL. First write the presentation: limit x tends to a, what is it? 2 positive. Convert to h: limit h tends to 0 f(2 + h). What was the formula? What was the formula? First it was RAL. Limit x tends to 2 positive f(x). RAL. Limit h tends to 0 2 + h. Here a is 2. Okay sir, so here came 2 + h. Now where are we going to put this 2 + h? In RAL. Where are we going to put it? In RAL. Look, limit h to 0, we will put here 2(2 + h) - 3. Now listen carefully, I am not doing this question, I am explaining the concept. Questions are a coming and going thing in maths. Now this question that has been formed, now this thing that has been formed, this has become your limit question, son, which you solved in 11th standard. In that, there was only one condition: if a 0/0 form is being made, if a 0/0 form is being made, then you cannot put the limit. So I will revise sandwich theorem, L'Hopital's rule, some formulas like sin x / x, don't worry. So you have to apply that. But here the denominator is one, son. Here the denominator is one. So you put it without hesitation. Zero can be formed in the numerator, right? 0/1 is allowed, right? But zero cannot be formed in the denominator. So now you put it without hesitation. Put zero in place of h. Directly put zero in place of h. 2(2) - 3. What was the answer, son? One. How much was f(2)? 7. And how much is our RAL, son? One. Did these two come out equal? No. So here we will write, since, since the function is discontinuous, discontinuous at x = 2. Or write it properly that, since, I wrote it wrong anyway. What did I write completely? Since our f(2) did not come equal to RAL, therefore the function is discontinuous, discontinuous at x = 2. Now what was asked in the question? Find all points of discontinuity. Meaning we are saying that it is not continuous at two. Besides that, everywhere else it will be continuous. You don't have to check. I am saying with certainty, the formula has been derived in such a way that it is a 100% reliable method. We can trust it. So this is one way to solve this question. It will take two questions to absorb the concept. In the third question, you will start running. Okay sir, after that you won't stop. After that, it's just questions, son. Look.
And look, this came. These are our NCERT questions. Now, for example, this question came, "not equal to." "Not equal to" means only one thing: both greater than zero and smaller than zero. Okay, look, for example, first I feel like doing this question because I need to explain that point again, so I'll come back to it now. Okay, let's do this one first again. Here the equals to sign came, so first of all, what will I find? One. In the question, again, "Find all points of discontinuity." We will find one, so 1 + 1, what was the answer, dear children? 2. Now what will I find? LHL or RAL? Tell me quickly in the comment section. Tell me quickly in the comment section. What did I tell you? This is equals to. Now we have to find the one opposite to it. Now, is the opposite one LHL or RAL? You tell me quickly. Is the opposite one LHL or RAL? Comment quickly. I hope you have done it. It's recorded, right? I can't see. So, brother, obviously, what came? LHL. I just told you, if the bird's beak is closed, it's LHL. If the bird's beak is closed, it's LHL. Limit, limit x tends to 1. You are checking at 1. 1 negative. Here came f(x). Limit h tends to 0 f(1 - h). Some children might be commenting, "Sir, in our school, they directly put it, they didn't do this H-thing." Agreed. But if you use the H-thing, when it was taught in your school, they must have said, "Beta, in this question, we will have to change the limit. So in this question, we will take theta or h. And in this question, there is no need to change the limit, so we will not take h here." A confusion arose in the child's mind: when to take H, when not to take H. What am I saying? Take it in all of them. Now, whenever a child is told, "Beta, apply this one formula and all questions will be done," the child becomes happy. So my teaching method says, brother, give one formula, all questions will be done. There is no need for bifurcation, whether to take H here or not. Those teachers will do some questions with H, some questions without H. I am saying, do all questions with H. It will take one
However, what came, son -1, then this function became continuous. This much must be clear to you by now. Okay? Now let's move on to some questions that require a bit of effort, where generally effort is needed. It came in 2012 and must have come many times, son, many times. Now, in this question, there will be a little trigonometry, a little log, a little exponential, and the value will increase slightly. Okay? Now the work will become a bit more effortful. The question asks: the function is continuous at pi/2. You need to find the value of k. Find the value of k. Okay? So, first of all, f(pi/2). What is the value of f(pi/2)? It's three. What is the value of pi/2? It's three. Okay? Now, take anything from above, like we took the RHL. I will directly put the limit h tends to 0. Please do one more step for formality. Limit h tends to 0, pi/2 + h. Look carefully, I will write big. k cos x. What did we put in place of x? pi/2 + h. Whole divided by pi - 2. pi/2 + h. Now, brother, we don't need trig. Don't say, sir, I don't know what cos(90 + theta) is. The answer is -sin(theta). You should know this, brother. If you don't know this, go and revise, son. It's going to be used in many questions. So, this came as -k sin(theta). cos(90 + theta) is -sin(theta). Second quadrant. It's not cos's house. 2 cancelled 2. pi cancelled pi. -2h came. Watch carefully. Look, brother, this omelet on the plate, h/h became 1. sin(h)/h became 1. So, what is the answer, brother? k/2. Minus cancelled minus. Correct? Okay? Now k/2 is left, and that is equal to 3. So, here again, you will write it nicely, with a bindi and lipstick, representing the question. Meaning, bindi and lipstick means makeup for the question. Since the function is continuous at x = pi/2, at x = pi/2, this function is continuous. Okay? So, will f(pi/2) be equal to RHL? That is, will 3 be equal to k/2? What is the value of k? It came out to be 6. And these questions are from NCERT. It's not like I've only brought NCERT questions. There are plenty, son, plenty. Wait, only half our work is done. I said, it will take at least 45 more minutes. You can see, I'm recording, I have an estimate. In front of you, it's written how long this lecture is. So, I keep continuity separate and derivative separate, every time, every time. Okay? So, the value of k is 6 here. Excellent question. Now, like this question, very easy question. It came in 2010. Do it quickly, quickly. I'm drinking water. Pause the video quickly and solve it yourself, son, yourself. Now, why do I keep saying this omelet thing repeatedly? Today I was at the gym. And I don't look like I go to the gym, do I? I have to tell people I'm going to the gym. My body is currently in a wrestling match, but I've reduced a lot from before. If any of you have been following me since 11th grade, I've reduced a lot. Anyway. So, today at the gym, the music stopped. An uncle was doing shoulder exercises. He said, "Yalla." So, "Yalla" came out of his mouth. Since then, I remembered, "Yalla, omelet." So, I've been saying this all day, all day. So, here too, look, a very lovely question. Okay? Now, here, first of all, we'll talk about f(2). So, k(2)^2, meaning the answer is, dear children, 4k. It's equal to this. Now, we'll talk about RHL. In RHL, there's no x. There's no x. 2 + h. So, the answer will directly be, son, three. Where will you put it? It will directly be three. So, since the function is continuous, these two will be equal. The value of k must have come to you as 3/4. You can say this question might have come in one mark, but obviously, it was the year 2010. So, it might have come in one mark only. Because if we talk about the pattern of 2010, it was 1 46. 2010 means 14 years ago from today. It's about a year or two after Ashu Sir, or maybe the same year. So, in 2010, the 1 46 pattern. It doesn't have the capacity for four marks. But at that time, you never knew anything. Anything could come. Anything could come. They used to give such big, such small questions for four marks. Look, this question was asked in 2011. kx. This is also an NCERT question. So, our NCERT is good. This chapter's NCERT is good. Okay? But some questions from Exemplar are very important. We'll do them now. x = pi. This function is continuous. You need to find the value of k. So, first of all, pi came. Put pi. k(pi) + 1. This came. Then comes, children, our RHL. Limit h tends to 0. Directly put h tends to 0. pi + h. Limit h tends to 0. cos(pi + h). Now, cos(180 + theta) is cos(180 + theta) is -cos(theta). Why minus? Because it came into the third quadrant. So, -cos(h). If you put 0, what is the answer, son? It's -1. The value of cos(0) is one. The answer is -1. Since the function is continuous, therefore, k(pi) + 1 = -1. So, pi k = -2. What is the value of k? It came out to be -2/pi. Just decorate the question a bit. Understood? Just decorate the question a bit. And look, like this question, it's an NCERT question. A question like this came in 2012. Not the same, or maybe it was the same. If it was like this, I would have written it. So, look here. Now, this function is continuous at both points. Meaning, it's continuous at 2 and also at 10. The question has become good. For now, NCERT is going on. Okay? So, first, I'll talk about 2. At x = 2. Look, x = 2 means f(2) is 5. Now, calculate this. This is RHL. RHL. Limit h tends to 0. f(2 + h). It became 2 + h. Put it here. Limit h tends to 0. ax + b. So, if you put 0, the answer is 2a + b. So, brother, since the function is continuous at x = 2, therefore, 5 is equal to 2a + b. First equation. 5 is equal to 2a + b. First equation. Correct? Okay? Now, if I talk about this question again, at x = 10. Check at 10. Now, at 10, the equals sign is here. So, the value of f(10) will be, brother, 21. What will it be? 21. Correct? Now, its opposite, 10, is RHL. The system must have been understood. RHL. Limit h tends to 0. This was also RHL, that was also RHL. Okay? Yes. So, here it came, 10 + h. So, where will you put it? Limit h tends to 0. Put it here. a(10 + h) + b. So, this came, brother, if you put 0, it's 10a + b. And then, since the function is continuous at x = 10, brother, 10a + b is equal to 21. First equation. Second equation. Can you solve it quickly? Tell me in the comment section, what is the value of a, and what is the value of b? Put a timestamp. It's been about an hour, right? So, put a timestamp quickly and write the values of a and b here. a and b can be solved by substitution. Clear? Done? Moving on. Next question. You can also do this easily. Look, what is the question? sin(x) - cos(x). If I calculate it, it's RHL. Limit h tends to 0. Directly put it here. It will come, son, sin(0 + h) = h. Divided by h + c(h). Now, it came in a recent exam. Okay? So, this question, which came in 2024, is very good. We need to verify continuity here, check it. So, first of all, the value of f(0) is 0. Then you can calculate anything. Like, RHL was calculated. Limit h tends to 0. Formula, write it down, son. 0 + h. Limit h tends to 0. h. Here it is h, and here it is, brother, sin(1/h). Now, if the value of h becomes 0, then 0 multiplied by anything, what is the answer? 0. Now, look at this beautiful question asked in your 2024. Okay? It was such a simple question. So, this probably came in two or three marks. So, here it came as 0. Now, similarly, what can you do? LHL. Limit h tends to 0. This is f(0 - h). So, limit h tends to 0. What is it? -h. And sin(-1/h). Now, this will also be, dear children, 0. So, all three came as 0. So, the function is continuous. The question is, but a bit awkward. Now, look at this. 2014, 17, 23. Look at the importance of the question. Now, observe this. 1 - cos. 1 - cos. This also came many times. I haven't written it, but look here. 1 - cos. Came many times. Look here. 1 - cos. Check, check. Oh, it's not here. Okay? So, brother, I've brought four or five questions. Otherwise, its variety is very large. So, now you look carefully. The content of these questions is quite important. I've written it in the first pointer in the member. Now, look how to crack it. You need to find the value of k. The function given is continuous. Correct? Okay? So, first of all, we'll talk about f(0). That's k. Now, not equal to 0. So, you can calculate anything. RHL. Limit h tends to 0. 0 + h. Directly put h. I've skipped a few steps. Obviously, those formality steps, you can do them yourself now. Okay? Because if we do every question like this, the video length will become very long, and more than half the children will look down. They won't even come for two and a half hours. Correct? So, this also needs to be considered, man. What to do? Limit h tends to 0. 1 - cos(2h). What was the formula? 2 sin^2(angle/2), meaning h. What is 1 - cos(2h)? Formula was written, son. 1 - cos(2h) is 2 sin^2(angle/2), meaning h. And what is below? Brother, 2h^2. 2 cancelled 2. Look carefully. This became sin(h)/h. The whole square. So, what is the answer? 1. sin(h)/h. What is its whole square? So, that's 1. 1 squared is 1. Now, look. So, therefore, with full bindi and lipstick, what is the value of k? It came out to be 1. Okay? Now, I tell you again and again, those children who didn't go beyond the one hour, 3 minutes, 4 minutes lecture, they've suffered a loss. Because the most excellent questions are starting now. Those heavy questions that you actually need, for which Exemplar is known, this topic is known for these questions. So, the upcoming questions, about 10 to 12 questions, those 10-12 questions are superbly important and a bit heavy. So, look at those questions now. They are starting after an hour, son. So, whoever left at the beginning will have trouble. Now, like this, it's the same format of question. If I talk about f(0), what does it come to? k. Just for practice, I brought this. h tends to 0. Directly put 0. h. 1 - cos(4h). And 8h^2. Can you apply the formula, son? Same question. This is 2 sin^2(angle/2). And here it is 8h^2. This will cancel out. It will become 4. Limit h tends to 0. Look carefully. This is sin(2h) / (2h). The whole square. The angle that should be above should be below. So, the answer is directly 1. Correct? So, therefore, here too, what is the value of k? It came out to be 1. The concept is very important, son. This function is very important. Now, look here. What is the value of k, son? You need to find it. Now, look carefully. In this type of question, you need to find the value of one variable, k. But you can find it through this and also through this. Now, the problem of choice is yours. Whether you find it from LHL or RHL. You'll get the same marks. You have to see which function you find easier. If you find the upper function easier, then the upper one. If you find the lower one easier, then the lower one. Okay? So, look. For example, I'm doing it from the upper one. What is the value of f(0)? Dear children, it's k. Now, limit. What is this? LHL. Limit h tends to 0. What is it? 0 - h. So, -h. And this is twice -h^2. Cos's minus is absorbed. And below, it's squared, so it will disappear. Limit h tends to 0. 1 - cos(2h). And here it is 2h^2. Again, apply the formula. Limit h tends to 0. 1 - cos(2h) is 2 sin^2(angle/2). Whole divided by h^2. 2h^2. This cancelled out. Now, sin^2(h)/h^2. This also became, son, 1. So, what is the value of k here too? It came out to be 1. And x/|x|. You just solved it. What was its answer? 1. Correct? Okay? Clear? Done? Let's move on. Now, look at this question. Here too, one value is missing, a. You can find it from this and also from this. Now, you've learned to find this. I'll show you how to find it from this. What is the value of f(0)? Son, it's a. Now, I'll find it from RHL. Because I've done this in three-four questions. Limit. You need to find the limit in the paper in only one way, son. Okay? Put a bindi. What came? 0 + h. So, here it came, sqrt(h) / (sqrt(16 + sqrt(h)) - 4). So, look. Now, here it's a fraction. If I put 0 directly, it becomes a 0/0 form, which is not allowed. So, this question needs to be fixed first. To fix it, what will we do? I'll count to three. Type quickly. One, two, spelling is a bit long. E, E, three. So, those who wrote "rationalization," "rationalization," very well written. So, what will we do, children? Rationalization. Okay? So, sqrt(h) / (sqrt(16 + sqrt(h)) - 4). So, here it came, sqrt(16 + sqrt(h)) + 4. And sqrt(16 + sqrt(h)) + 4. Brother, if there's a minus here, you multiply and divide by plus. Limit. Now, now children will say, sir, we'll do it from the top. Forget it, what's the hassle? But it's my duty to teach you how to do it if someone wants to do it from here. This came, sqrt(h) * (sqrt(16 + sqrt(h)) + 4). And below, the formula applies. (a - b)(a + b). So, the square of the first minus the square of the second. 16 cancelled 16. And sqrt(h) cancelled sqrt(h). What remained? sqrt(h) tends to 0. 16 + sqrt(h) + 4. Now, you can put 0 in place of h. This will be 4 + 4 = 8. So, brother, what is the value of a? It came out to be 8. Okay? See, gradually, gradually, the level of the question is becoming excellent. Becoming excellent. It's important. You can see it came. In compartment, All India 2010 compartment. Okay? Now, here it's just written, came many times. Look, in Shivdas's book, it's written for only one year. But after reading the whole chapter from Shivdas, you'll see that the same question, a similar question, the same type of question, maybe k came in place of a, or m came. So, it came before. Okay? So, it's multiple times later. But yes, it's not written anywhere in Shivdas that the same question is written multiple times. So, all the years are written in brackets. They've written only one year in brackets. But I can confidently say this question came many times because I've seen so many question papers in my life. Now, look at this question. Find the value of a. You need to find the value of a. The function is continuous at 0. So, how much practice have I given you? 1 came. Take RHL. You can take anything. Not equal to 0. Limit h tends to 0. What came, brother? sin^2(a) / h^2. Now, look. The angle that is above, you need the same angle below. The angle that is above, you need it below. You'll have to create it. Do you remember 11th standard a little? This is, brother, sin(a) / h^2. What will we do now? We'll create the angle here. Multiply by a, divide by a. Now, this became 1. And here it remains a, and its square, meaning a^2. Now, these two are equal. a^2 = 1. So, what is the value of a? It came out to be plus minus 1. Correct? Clear? Children? In this way, these questions are becoming better and better. And look, like this, it's a very lovely question. The formula for exponential and log also got applied here. Look, look, look. Here we need to check continuity. Check continuity. Is this function continuous or not? Show it. With the blessings of Mother Goddess. First of all, first of all, talk about f(0). What is the value of f(0)? Brother, 3/2. This is our LHL. All three are given separately. Even to identify them. You can calculate anything. h tends to 0. Directly this came, sin(3 - h) * tan(2 - h). Because if you put 0 - h, then here it came, limit h tends to 0. So, -sin(3h) and -tan(3h). Below it's 2. Now, sin(h)/h is 1. You need to create the angle that is not there. It's 3h above. What will you do? Multiply by 3h, divide by 3h. Here, multiply by 2h, divide by 2h. This formula also exists, son. It's 1. And this formula also exists. What is it? 1. Minus cancelled minus. What remained? h cancelled h. Directly the answer is, son, 3/2. Directly the answer is 3/2. So, multiply by the same angle, divide by the same angle. Now, if I take RHL, limit h tends to 0. Look here. It's log(1 + 3h) / (e^(2h) - 1). Again, you'll have to do the same thing. log(1 + x)/x should be there, right? So, I'll multiply by this, divide by this. e^x - 1 formula was written. Go back and see. So, e^x - 1 / x, that also becomes 1. Limit h tends to 0. log(1 + 3h). Multiply by 3h, divide by 3h. e^(2h) - 1. Multiply by 2h, divide by 2h. Dear children, this also became our 1. Formula. This also became our 1. Point to remember. h cancelled h. This also came out to be 3/2. So, all three are 3/2. So, this function is continuous. This is an Exemplar question. Many questions in this are from Exemplar that have come in past years. You can say if this hasn't come, then it's most expected. You can see how beautiful the question is. Three formulas are applied. From 11th standard. Then this question came. Look how beautiful the question is. This came in 2016. Okay? Complex question. A bit complicated. Five marks. Four or five. Out of which one mark is for one or two MCQs. Okay? The first four-five questions you have are MCQs of that type. When I bring the series of extra questions, I'll do them there too. Otherwise, they don't come much from continuity. They will be direct, or one of these. Or direct. Brother, x^2. Tell me, is it continuous and differentiable or not? So, when we study differentiability, you'll understand. Then one or two marks, one three marks. Meaning, either a three-mark or a two-mark question of this type. So, its probability of coming has decreased now. Okay? Firstly, there might be something wrong with the question. So, here it should be bx^2. Here it should be bx^2. It should be x^2, not b^2. bx^2. Okay? Anyway, if we talk, then f(0). What does it come to? c. Now, we'll talk about LHL. Limit h tends to 0. sin(a + 1) - h. And below it is -h. Common minus from all three and cancel it, son. Limit h tends to 0. sin(a + 1)h + sin(h). I'll write h separately below both. Okay? Now, I need to make this a bit smaller, brother. A bit smaller. Correct? Look here. This came, limit h tends to 0. sin(a + 1)h. Multiply by a + 1h, divide by a + 1h. So that the same angle is formed. And here, the angle was already the same. So, we made it 1. And this also became, son, our 1. What remained? Limit h tends to 0. a + 1 from here. And 1 from here. So, the answer is directly a + 2. Directly, dear children, a + 2. Clear? This came, a + 1. And this, this became 1. a + 1 + 1 = a + 2. Now, let's talk about RHL. This is a bit important. So, RHL. Limit h tends to 0. So, sqrt(h + b^2) - sqrt(h). Divided by b h^(3/2). Correct? This became, meaning 0 + h. So, I put h in place of a. Now, look carefully. It's becoming a 0/0 form. So, I can't put h directly. Nor is any formula forming. What will we do? Common sqrt(h) from above. What remains? 1 + b*h - 1. And below, 3/2 means what? h * sqrt(h). sqrt(h) cancelled sqrt(h). Limit h tends to 0. sqrt(1 + b*h) - 1. Divided by b*h. Now, again, some big spelling, rationalization. Because it's still a 0/0 form. Many children, even after 12th grade, think rationalization is only for the denominator. Okay? I am very sorry for the disturbance. Yes. So, here it came, sqrt(1 + b*h) + 1. And sqrt(1 + b*h) + 1. Brother, a + b, a - b. So, a^2 - 1^2. It's 1. This came, b. Below, this remained the same. And look carefully. 1 cancelled 1. b cancelled b. What remained? Look carefully. 1 upon. Now, a very good conclusion is going to arise. If I put h = 0, the answer is sqrt(1) + 1 = 1/2. Come here, brother. You needed to find the value of b. Look, b disappeared. I needed to find the value of b. b disappeared. c, a + 2. It's equal to 1/2. How will b come? Let's tell you. First of all, c is 1/2. c is 1/2. a + 2 is 1/2. So, a is -3/2. Tell me quickly in the comment section, what will be the value of b? Tell me quickly in the comment section, what will be the value of b? I'll count to three. One, two, three. So, the answer is, b could be any real number except 0. Look, if you put anything in place of b in this function, say 5 lakh, 5 crore, sqrt(5), 20 crore, 20 million, 20 million dollars, anything. b will disappear. It won't become 0. b will disappear. b will disappear. Whatever I put in place of b, multiplied by a, it will become 0. But I can't put 0. Because if I put b = 0, the whole function will be undefined. So, b could be any real number except 0. Now, tell me, what a beautiful conclusion came. I'm saying that if the author, meaning the one making your question, doesn't give even one question from this topic in MCQs and directly gives a four-mark question. It's become a beautiful question. Beautiful question. So, it's most expected. And it's not like it hasn't come. It was asked in 2016 for four marks. Okay? So, now the story is, now you have to decide yourself what the importance is. Such questions are nowhere in NCERT. Now, look at this dangerous question. A child will leave half of it. You need to find the values of a and b. But it's easy. A little trig and a few points to remember from the slide should be well-known. Now, look here. First of all, f(pi/2). What is the value of f(pi/2)? Son, it's a. This is LHL. Obviously, this will be our RHL. You can calculate anything. Tell me, we need to find the values of a and b. So, first, I'll calculate LHL. Because we have to calculate all three. Limit h tends to 0. This came, 1 - sin^3(-h). And below it is 3 cos^2(-h). Minus will come out and become plus. Here it is 1 + sin^3(h). Divided by 3 cos^2(h). Now, 1 - tan. Oh ho. I am very sorry. Very sorry. I know you've caught the mistake. The limit is not 0, right? It will be pi/2 - h. The result of haste. This came, 1 - sin^3. Not -h. It is a - h. a is pi/2. And this is 3 cos^2(pi/2 - h). My bad. Okay? So, here it came, sin(90 - theta) is cos(theta). So, 1 - cos^3(h). cos(90 - theta) is sin(theta). 3 sin^2(h). Below, let's do the same thing. Multiply by h^2, divide by h^2. This became 1. Limit h tends to 0. 1 - cos^3(h). Divided by 3h^2. Okay? Look carefully. Now, this is becoming a 0/0 form. What is this becoming? 0/0 form. Now, many children, who often have doubts, I know you also have them. Sir, write it like this. Where will the cube go? Sir, let's do this. Here it was h^3. Look, 1 - cos^3(h). So, let's make it h^3 here too. Let's multiply by h. So, this will become 0. Son, the 0/0 form is still not over. Check it. The 0/0 form is still not over. And anyway, that function, it should be 1 - whole cube, right? 1 - cos(h) whole cube, right? Not 1 - cos^3(h). The formula is not applying. So, here we have to apply the formula a^3 - b^3. Limit h tends to 0. a - b, a^2 + b^2 + ab. We applied the formula a^3 - b^3. Okay? Which was learned in 9th standard. Here it came, 3h^2. Limit h tends to 0. 1 - cos(theta). Point to remember. 2 sin^2(angle/2). And keep this as it is. Put 3 below this. And put h^2 below this. Look carefully. Limit h tends to 0. 2 sin(h/2). Divided by 3h^2. Okay? Look carefully. Limit h tends to 0. 2 sin(h/2). Divided by 3h^2.
H whole square, 1 plus cos square A plus cos A divided by 3. What will you do? Multiply by 2, divide by 2, so that this, my child, becomes one. What remained? 2. What remained here? 1/2 whole square. Here 1 plus 1 plus 1, because cos 0's value is 1. So here also 1 plus 1 plus 1, divided by 3. 3 cancelled by 3. 2 divided by 4. Answer is 1/2.
If not understood in one go, then go back and rewind. 1/2 came, our value, my child, A's value. Now B has to be solved, this. So that means now you will have to find RHL. Find RHL quickly. So what is RHL? Limit h tends to 0, and we will put here b (1 minus sin (90 plus h)). This was put, 90 plus h, and here came pi minus 2 (90 plus h) whole square. See, wherever x was, there I put 90 plus h. Now the formula, sin (90 plus theta) is cos theta, it is positive only. So limit h tends to 0, b (1 minus cos theta). Below, 2 cancelled by 2. Pi cancelled by pi. Minus 2h square remained. Above, again the same formula came, dear children. What came? 2 sin square (angle's half). And below came 4h square. Cut it. Limit h tends to 0, b by 2. This came (sin (h by 2) upon h) whole square. What will you do? Again the same, multiply by 2, divide by 2, to make the same angle. This became, dear children, our one. Now b by 2 into 1/4. So what is the value? b by 8.
Now this function was continuous, so A and here came 1/2. A equals 1/2 equals B by 8. So A's value came half, and half, which is, is equal to B by 8. So B's value came four. Isn't it a blockbuster question? Do it twice, brother, this. Do it twice, please. The point is understood. This is the last question. This, you will do. Find the value of A. This is in homework. Tell A's value. It is in homework. Tell A's value.
Now you can do any question from your NCERT comfortably. Any question you can do comfortably. Now, as I leave, I will take two minutes and tell you one thing, that suppose, put this also in "point to remember." Point to remember. Now, the last questions that you are seeing, the very last four or five questions, they are cos x square. See, there is something called composition of functions. It doesn't come that much because this composition of functions has been removed from relation and function. But I had taught you a little bit about it in both 11th standard and 12th standard. Composition means, suppose there is one function f(x) and one function g(x). Okay, sir. They both are continuous at x equals C. Both of them are continuous at C. So therefore, f plus g, f minus g, fg, f divided by g, all these will also become continuous at x equals to that critical point. So the last question that you must be seeing, it must be cos x square. It says, check if this is continuous on real numbers. Yes, see, this x square, what is it? It's a polynomial function, it used to be continuous. Cos, which used to be, what used to be? Continuous used to be. So this also what will be? Continuous. Now, for example, if something like x plus mod (x plus 1) came. What is this, my child? Module, and what is this? Linear. This also what will be? Continuous will be. So directly tick, tick, tick, keep this in mind. The point is understood. Okay, sir. The rest of such questions we will do in MCQs. I am not refusing. But today's target was not MCQs. Why was it not? Because they are asked less. And if they are asked, they are asked with differentiability. And differentiability is a topic of derivatives. It will be with the help of continuity only. Okay, sir. Meaning, you will have to sharpen this topic, then only you will be able to read the next topic, differentiability and derivatives.
Revise derivatives of 11th standard. In my 11th standard, you will have two links for Limits and Derivatives. You will go and see, one will be Limits. Like here, in this chapter, what came in the beginning? Continuity was big, derivative was small. Similarly, in 11th also, Limits was big, and derivative was small. So that is the content of Limits. If derivative looks big, then you must watch that lecture before the next lecture and sit. You will get a lot of help. Because it's obvious, I will tell all those topics of derivatives from 11th. But still, if you don't know how to take the derivative of x square, x cube, cos x, sin x, I kept the basic things aside first, and then I did concept, concept, concept, concept, like I did here. So that lecture is very important for you. So those whose derivatives are weak, they will come here after watching the derivative of 11th standard. Even if you just watch, you will understand, don't even do anything. Okay, right? And then derivative will come. Okay, lecture, absolutely, if you have understood, if you have received a lot of help, then please tell in comments, give your feedback, and some positivity note, an emoji. If you got help, like this event. Do share with friends, because if you do good, good will happen to you. I will meet you in the next lecture, one shot of derivatives. Until then, bye bye, take care, have a nice day.