continuity part 3 maths grade 12 cbse

追加: 2026-05-19

[00:00:02]I like the computer.

[00:00:38]So what you have to do here is you have to find continuity. Now you know how to check the continuity. How to go for checking continuity left hand and right hand continuity. Now let's see how we are going to see modus questions. I have given yesterday and last class I have discussed about modus question and I have also discussed like if we want to check continuity for modulus we are finding that the line one increasing line one decreasing line will be there understood or not so two different modulus you have to check the continuity sum of these two different modulus continuity if you have to check how you will be checking correct now what if if three moders will be given what If if three modulus will be given.

[00:01:23]Good morning.

[00:01:26]Good morning.

[00:01:31]Good morning everyone. So now look at what if if three modus will be given at that case when three modus will be given how you are going to check the continuity. Good morning. So now look here. So first is somebody will be giving you to check the continuity means you can start with f of x is equal to modulus of x x -1 and x - 2. Now solve it and find the continuity. Okay. If three modulus will be given 1 2 3 back to back three moduluses are given that particular time how you are going to check the continuity for this three modulus beta whenever remember only one thing all the continuity what you are finding you have to find in its limit critical point understood not so whenever they will give to check the continuity they will give to check the continuity at x =0 x = 1 and X = 2 that means here modulus of X part internal X part make it equal to zero and here X -1 make X -1 =0 and X -2 make X - 2 =0 that means how many points you are getting you are getting three points. So here they will not tell you to check the continuity only for two point. Here they will tell you to check the continuity for three point or they will give you some other values to check the continuity. So this is exactly the way they ask question.

[00:03:04]Are you understanding? So maybe they will give you two critical points to check the continuity. See there is no other point to check what modulus part you are going to give because you know that modulus is made with two lines. One is decreasing and one is increasing.

[00:03:20]Understood or not? So in between if you will put anything here any number here 100% left side and right side is continuous. So where they are going to check the continuity is the point where it is increasing and decreasing and increasing. Understood? This will become the checkpoint for checking continuity.

[00:03:39]I hope everybody understood. Sis are you there in the group? Nishika, Pioja, Sugata, Kushi.

[00:03:48]Come on, check this one out. And this is most important. So now we will check the continuity at x =0, at x =0, x = 1 and x = 2. Until now we got a very important information about Sidesh. Good morning. We got a very important point about continuity and that important point is it is continuous. Understood or not? Whenever in modus we are taking critical point we are checking it is always coming as continuous. Understood?

[00:04:24]So now in examination now if they will tell you to check for it continuity then we don't have to do very big work. We can directly say that it is continuous.

[00:04:34]If modulus is involved in it, we can directly say that it is continuous.

[00:04:39]Understood or not? Kushi, are you there?

[00:04:41]Nishika, are you there? So now look here. So now what we are going to do here is we are going to respond to the question. Are you ready for it?

[00:04:50]Children, are you ready for it? Come on.

[00:04:57]Okay.

[00:05:27]Understood? So this is the thing we have to do. Already I have taught you how to do it. No big work. What we are going to start with. First our work is to draw the line x =0, x = 1 and x = 2. Understood? So here three points will be there to check 0 1 and two and after that what next work you have to do. First here less than zero here x will be greater than zero and less than one. Here x will be greater than one and less than 2 and x will be greater equals to. So these are how many critical points you are going to check. Here this part is x is greater than and less than. So this is 1 2 3 4 four zones are there and every critical zones you will be getting some values. I hope you might have understood but LHL and RHL they both will be coming same.

[00:06:25]So now here for f of x see this is the most important part and here only f of x part you are going to make it nicely.

[00:06:33]Now look here I told you always when continuity you are checking first you have to break modulus and the most important thing is break modulus before solving break modulus before solving don't break modulus after putting the question are you understanding or not after putting the question is after putting in LHL better and the most convenient way is to solve it here only then LHL RHL will become very easy to diagnose understood or not? So let's talk about it. Let's see how we can do here is first is when we are going to take value x less than zero. So first our point is this given value x is coming lesser than zero. So any value lesser than zero. Let's take any value lesser than zero. Any value lesser than zero means negative. Understood? If I will put here negative means inside x value is negative. That means modulus will open with positive. Understood? Or not? But inside the value is negative. So what we will do? We will keep the bracket outside is negative. Means it is now modus is in negative zone. Understood? So here means the less than one means the modulus is in negative part. Here also less than one. Less than one means any value you take already here minus1 is there. So if I'm taking minus1 minus1 minus1 will become minus2. So 100% this part will also be negative. So it will become x - one. This part again because this is the biggest number x - 2. So this will also become min - x - 2. So when x will be less than zero all the three part of the modulus will be negative. Understood? So now next part given x will be lying between -1 to 0 to 1. 0 to 1 means any value between 0 to one you substitute and check. So if you are checking any value between 0 to 1 you can take 0.5.

[00:08:34]Between 0 to 1 we can take 0.5. So 0.5 if you're substituting here mean it's a positive number. So modus will open with plus. Now here if you are putting 0.5 then 100% this is negative and this part will also be negative. So this will open minus and this will open minus. when when the region is x is greater= 0 and less equals 1. So this is the second case. Now third case attendance giving again and again I'm calling no people are telling the attendance.

[00:09:08]Next part is between 1 to two between 1 to two means next time we are going to take 1.5 can we take 1.5 between 1 to two if we are taking 1.5. So 1.5 means this one will become positive. 1.5 means 1.5 minus 1 positive. But here 1.5 minus 2 this will ultimately become negative.

[00:09:32]So here this part will become positive positive and this part will become negative when when x will be greater than one and less than two. Understood?

[00:09:44]So one two three zones we have divided.

[00:09:47]Three zones we have divided. The last zone is going to come for this part when x is greater equals to. So when x is greater equals to any value we will put inside modulus inside modulus values will start coming negative or positive positive. So here if I'm taking x greater than 2 mean this three. So if I'm substituting here three so this is positive 3 - 1 2 this is positive 3 - 2 1 all positive. So this time it will open all positive. Let's check it will become x + x -1 + x - 2 when x is greater equals to 110% done. So this was the zone division. So once we have divided the four part okay this this are there now solving will become easy.

[00:10:39]Correct? So this was our first work. We did our first work. Now we have to check continuity. So one by one individually check continuity. Now the first part to check continuity is this section continuity at x =0.

[00:10:56]Continuity at x =0. We will check continuity at x =0. This is the point.

[00:11:02]First we are going to check continuity for this. Okay. So if you are checking continuity for x=0 what work you can do. First LHL and RHL systematic approach again we are going to check. we will write here continuity at x =0. So now for continuity at x=0 only dedicate yourself to x=0.

[00:11:30]So now you check what is in left of 0 and what is in right of 0. Only this thing you have to check. Okay. So what can be left of zero? Left of 0 is x less than 0. X less than Z left of zero.

[00:11:46]Understood. What is in right of zero?

[00:11:49]So this part is coming. That means for checking continuity for X =0 we need to take these two only only these two and the first part will become LHL and the second part will become RHL. That's it.

[00:12:04]The first part will become LHL and the second part will become RHL. Easy and to be understandable. easily understandable or not. So what we are going to do here?

[00:12:15]We will write LH L at X =0. After that what is our next step? Limit X ts to 0 - f ofx means we are going to which zone we are going to go in the left part.

[00:12:32]Understood not. So which is less than zero. Not exactly less than zero.

[00:12:36]Tending to be less than zero. So after that what function we are going to write? Come on, we will make it small and we will write here. So now look here this part is - x - x - x if I'm resolving it - x here it is x -1 - x - 2 open it what you are going to get - x - x + 1 - x + 2 - - 3x + 3 what the value is coming - 3x + 3 is it coming continuous?

[00:13:09]It will come without solving now in exam now without solving this MCQ don't be overconfident there okay because a little bit clause here and there they will put and check so you have to be very careful when you are reading the question two times three times first you read the question if any terms and condition given means then you put the terms and condition and check so now look here first - 3x + 3 what is written here - 3x + 3 so what we will Let x = 0 - h since x ts to 0 minus therefore h ts to 0 for all h is greater than equals z.

[00:13:49]So h is a very near number a number which is a number which is very very near a number which is very very near.

[00:13:57]So let's talk about this number z h. So this time we are going to take limit h x ts to0 instead of x. Now what we are going to substitute 0 - x + 3 then so wherever x is there now we are going to substitute what we have assumed our assumption was x = 0 - h so you'll write 0 - h now h is tending to 0 so it will become 0 - 3 + 3 will come as 0 so LHL is coming as 0 now what about RL another value we are going to write RHL L at x =0. So what we are going to write limit x ts to 0 + f of x and this time which one we are going to take this time this is coming as - 3x + 3 correct after dissolving and here after solving what you are getting x - x + 1 - x sorry here it is - x second one also it is coming first one is plus so - x + 2 so + x - x cancel - x + 3 this time For this one value is coming - x + 3. So this time we are going to take - x + 3. RL let's see it is going to work or not. So here limit x ts to 0 + - x + 3. Till here is it clear?

[00:15:29]Till here is it clear? So now we did nothing. First understand when modus question given this part equal to zero this part equal to 0 and this part equal to zero three critical points we got 0 1 and two we have to check continuity at 0 1 and two no other point okay after that what we did we took the regions less than 0 greater than 0 less than one greater than one after that less than two greater than two so automatically the numbers will jump like that only after that what next work we have done whenever value lesser than zero will be greater so modulus will open with negative because all negative values we are taking now so all negative after that second part between 1 to two sorry 0 to 1 0.5 first one is coming positive so the negative between 1 to two 1 2 is coming positive third one is coming as negative more than two all the three are coming positive till here then after that we are checking LHL RHL at X=0 the first checking point and for this what we have seen we are checking continuity at X=0 so LHL and RL part we are taking so when LHL and RHL part we are taking we are seeing that nicely the values are coming in our hand correct now so now what we are going to do with the next part we are going to write let x = 0 + h since x ts to 0 + therefore h ts to0 0 for all hes greater equals 0. So here we will write limit x ts to 0 + here - 0 + h + 3 is it possible because here when you are solving it is coming for what? x - x + 1 - x + 2. This got cancel - x + 3 is there. Huh?

[00:17:42]Huh? Plus so plus. So here it will become three.

[00:17:48]That is not zero because plus three is separate. So here this part will multiply with zero. Here I thought this part directly I did. So three is there.

[00:17:56]So LHL and RHL both are coming as three.

[00:17:59]Yes, answer is correct. So three will come. So LHL and RHL is coming as three.

[00:18:04]So once LHL and RHL is coming three means center also point if we are taking directly for continuity only for zero the answer will come three only. That means it is continuous without solving also by seeing the question and seeing the critical point we can say that it is going to be continuous. Modulus is proving continuity. So we can say like that yes absolutely. So now can we go for the next question please?

[00:18:29]Will you be able to check the continuity for x= 1 also and 2 also for x= 1 which one will be the left part and which one will be the light right part? Can you tell me for checking continuity at x = 1? So if somebody will ask you to check continuity at x = 1. So I'm checking continuity at x = 1. So here this one and this part we will take correct now.

[00:18:55]So for x = 1 means this will become my LHL and this will become my RHL. This time the first one I'm not going to take. Understood? For checking continuity at x =0 what we have taken LHL RHL for checking continuity for X= 1 LHL RHL for taking continuity for X=2 LHL and RHL are you understanding so this is the division I hope everybody understood yes so now we are going to solve one question are you ready for trigonometric some questions now we must solve which is exactly little bit tricky Okay, write a question.

[00:19:47]Okay, a signum function is continuous or not. A simple question I have asked without solving signum function.

[00:19:57]You know what is signum function? If somebody will write modulus of x by x it will be coming under sigum function.

[00:20:05]Mostly the range will be coming as minus1 0 and one. Graph also we have discussed signum function is going to be continuous function. Okay. Uh why can't we prove it? So if it is continuous means we should prove it now. Okay.

[00:20:22]Taking the risk proving it. Is it going to be continuous? By graph it is showing something by mouth it is coming different. So let's see specifically if somebody will tell you signum function for because see simple part if somebody will be asking you mean miss don't judge so fast. So here I will be solving signum function mean I will definitely putting one circle here and then I'll do like this specifically for this and then I will do a circle here and then I will do like this. So 100% I'm not going to touch the center value.

[00:20:59]So and if very person very intelligent person will say that he check continuity at x=0 and you know what where x=0 on yaxis where is x =0 on yaxis come on check continuity at x =0 go for it check continity at x=0 for do it what without solving we can say now by seeing graph itself we can say they're telling to check at x=0 go left of it all negative value minus one you check without solving also we can have brain now so when x value will be less than zero come on divide it into two regions x =0 they gave us the critical point as x =0 go for x less than0 and go for x greater= 0 that's it when x less than zero means f of x x will not change but modus part will become negative lesser zone x negative modulus part only is going to reflect negative so it will become minus x variable will remain same so here without solving answer is coming as minus1 now minus1 is fixed. Correct?

[00:22:25]So whatever value you take x = -3, x = -4, x = -1 lakh also. What will be the value of y? Minus one only. Correct? Now because here you put in modus section you put here they told lesser part.

[00:22:41]Lesser part means modus will open with negative. Understood. And here for more part is x greater than zero means it will come x by x. So no matter what value you are putting it will always come as one. I hope everybody understood. So this is one very important thing we must remember. So without solving also the things we can capture. So left side is minus one. Right side is always + one and you are standing in middle. Got it?

[00:23:12]You are standing in middle. So 0 by zero form is already not defined. So good morning Sakshi. 0 by 0 is already not defined. 0 by 0 is never defined.

[00:23:24]Correct? Now because 0 by 0 when somebody will give you you will say 0 into 1 by 0 and 1 by 0 is infinity. So it will become infinity.

[00:23:35]Are you understanding?

[00:23:37]Simple concept. Yeah. Don't exaggerate it more. Don't make it more critical.

[00:23:42]Just understand this one. So zero will not be test. Is it continuous? No. So signum function is never continuous at critical point understood without solving in examination you can't diagnose everything some things you have to perceive from the beginning and this is the only thing function modulus is there with variable and denominator means you need to go with that phase so modulus of x by x definitely it is going to be this side minus one that side plus one so signum function is continuous or discontinuous signum function is continuous or discontinuous discontinuous at critical point the point see if they will give you if you are like they will give you any point from here continuous if they will give you any point from here continuous are you understanding like a simple part I'm giving you you just understand the function given to you is nishika siddesh everybody sakshi please kindly check this one out if you are there please just understand this one f of x is equal modulus of x by x you have to check continuity at x =0 100% it is discontinuous critical point already we have drawn the graph but same question if they can frame like this now you are knowing it is discontinuous your brain is working as it is discontinuous but they will tell you to check continuity at x = - 5 then what you will do solve key now specifically we are dealing with -5 x = - 5 what x= -5 come on use your brain first what left of -5 right of minus 5 anything you do minus1 and minus1 only for because minus one is fixed. Now minus one is fixed. Now you will say auntie can you solve it one time.

[00:25:59]So if somebody will tell you because in left side so here x is less than so we will be planning it very carefully. Plan should be very careful. Less than part also minus y will be there. Right then part also minus 5 will be there. How left side and right side will be minus 5 that also should come in your braid. You should think about that. Correct. Now less than -5 and greater than - 5. Less than - 5 we will take - 6.

[00:26:28]Think let's see any cost is coming when we are thinking. No at least we are understanding something. Is it going to be continuous or discontinuous? When we are not taking critical point, we're taking any point away from it. Already we know signum function. Signum function is telling one thing a go like this for me. This is minus1 and this is one.

[00:26:51]Correct? Now we have taken a critical point - here. Now left of minus 5 and right ofus you will check it is going to be continuous or not. So x less than - x greater= - 5 x less than - 5 means negative greater than greater than.

[00:27:23]Okay.

[00:27:26]Done. Very very nice. Smile came. So when we are unable to when we are unable to solve means we will only smile. Nice. That's a good strategy. About you madam online children.

[00:27:43]If good morning good morning to solve x less than - x= - x less than - 5 means what value you will take - 6.

[00:27:59]- 6. So -6 mean modus left side will be open - 6 6 by - 6 -1 will come fixed correct here also when you are writing greater than - 5 greater than - can I take -4 so modus inside will be there because all are negative all are negative zone then critical point correct now so it will come like -4 again the value is coming as minus one continuous the line is a continuous part unders or So when line is a continuous part is minus1 minus one. So away from critical point left side right side values are same only coming no other thing is coming.

[00:28:42]I hope you might have understood this one. So here left side and right side both are giving you minus one by for this minus one the value is coming as minus one. And for this point if you will take five the value is coming left side also one and right side also one.

[00:28:59]So definitely it will be so they will not be telling you to check continuity at these things understood they will tell you to check continuity at the point where it is going to be continuous critical point and critical point was zero there I hope you might have understood okay now one question I'm giving without solving tell me without solving no need to touch the question and solve it now Again critical point they will give you zero at center point it is zero without solving is it continuous and discontinuous just very good so without solving continuous discontinuous let's see how many of you can say the answer without solving just till it is continuous and discontinuous they changed the critical point did you saw that when you were doing modulus of x by x they told to check at x= to0 that means this part they're taking it equal to zero so they told to check the continuity at x= 0 now where they are sending you they're sending you this part make it equal to 0 x - 4 =0 so now they're telling you to check for x = 4 now go for in your brain. Now something should come. Think see absolute absolute without solving. Yeah. One confusion is enough for us. Same function is coming. What did I said? The modulus of x by x same is coming. It is going to be in signum function format.

[00:31:01]Here also it is in signum function format. Modulus of x - 4 by x - 4. So what we going are we going to waste time specifically when this story is given already? We know that one thing you just do first you check one thing.

[00:31:17]Take this one out.

[00:31:20]X = 4. Center value what they have given zero. Now X less than 4 that means upper side modulus will open with minus sign.

[00:31:34]What value is coming? Minus one. Now here x greater= 4 strictly I'm putting greater than this one will sorry opened with plus and this one is also opening with plus answer is coming as one so without solving we can have this type of answer left side is minus one right side is + one center is zero how can it be continuous again it is showing which function sigum function where range is min -1 0 and one only continuity.

[00:32:15]The chapter is continuity.

[00:32:18]Did you got this? Did you understood this? That is -1, one and zero. Simple as it is. So is it going to be continuous or discontinuous? Now at least answer it is going to be online children. What it is going to be after sleeping? If anybody woke up, please answer the question.

[00:32:49]It is not absolutely correct. It is not going to be continuous. It is discontinuous.

[00:32:55]I hope understood. H. So now this is exactly the most important part.

[00:32:59]Continuity, discontinuity, it'll be checked like this. So any questions like this is given to you. What answer you are going to say? It is going to be discontinuous. But you will say this is continuity different maths continuity and defensibility. Now can you answer my question? So here if they will tell you by solve how to solve it left side is coming as minus1, right side is coming as + one. So what you will write? Checking continuity at x = 4 l at x = 4. So what you are going to write limit x ts to 4 - fx are you going to do very big here? Anything x t 4 minus answer is minus one. Answer is minus one. Without solving here, hn thing is not coming correct. Right side what you are going to write hn at x = 4.

[00:34:06]Limit x ts to 4 + f of x and here it will come limit x ts to 4 + 1 answer is coming as 1. So LHL is not equal to RL.

[00:34:18]That's it. LHL is not equal to RHL as simple as it is because here already so only I'm telling you you make this line half of the problem will be solved understood not so strategy ready means you will be able to solve without any pain correct so now here our strategy was ready so LHL at X = 4 is not equal to RL at X= 4 any other pain is there no pain Huh? Understood? Nicely.

[00:34:55]Okay, let's go for it. Sin theta continuity everybody knows. Cos theta continuity everybody knows. So, are we ready for solving a question where continuity will be checked for sin x by x? Come on, solve this question.

[00:35:23]Any intelligent person tell me f of x sorry x - modulus of x all modus section question first solve by 2 given when x is not = 0.

[00:35:58]Next part is two when x = 0. Come on, go for it.

[00:36:06]Again and again, I told you all the tricks.

[00:36:12]It's not for 11, it's for 12.

[00:36:33]again. First draw the line. Most of the story will be over when you will draw the line. Every time now draw the line and solve your line drawing will save you from all the critical conditions.

[00:36:49]All after put drawing the line putting model Sesh Mishika Pushi Sakshi.

[00:37:34]Can I solve M less than part.

[00:38:00]Most of the time I told you when x =0 is there, they give the value two.

[00:38:10]Okay. Now here x not equal to 0. x not equal to0 means either less than 0 or greater than 0. So exactly not equal to 0 but it can be less than 0 and greater than z. So what you are going to do here x less than 0 and here what is there? x is greater than 0. So when less than 0 will be there modulus will open with negative sign. So here x will be as it is but modulus will become min - x by 2.

[00:38:42]So it will become x + x by 2. It will become 2x2.

[00:38:48]So it will come as x.

[00:38:52]Understood? Now go for this side. So when it will be open f of x is equal to greater than this positive part. So for positive part modulus will open with plus. So x - xx2 answer is coming as zero. Here only story over. Yes absolutely correct. So here center value is two. What is your center value?

[00:39:16]Central value is coming as two. Right hand side is coming as zero. So you can equate center and right and say that it is not continuous. Correct. By checking these two point itself we can say it is not going to be continuous. Absolutely correct. So now here this is exactly the most important point to be judged. So here write continuity at x= 0 LHL at x =0. So what you'll write limit x ts to 0 minus f of x and what is the number given for you is limit x ts to 0 minus it is coming as x understood? So here it is coming as 0 only because the moment let x= 0us h since x ts to 0us therefore h ts to zero for all h is greater equals z. So what value will come limit h ts to0 it is 0 - h answer is coming as 0.

[00:40:21]So right side is coming as 0. Ll part is what?

[00:40:28]RHL at X =0 limit X ts to 0 + f of x is equal to limit x ts to 0 + 0 is coming answer is coming as 0. So left side and right side both are very obedient and they're putting zero. But what about the center? In center they have given value as two. So LHL is equal to RL but continuity says that LHL RHL should be equal center value should also be equal but in your LHL part is coming as zero.

[00:41:03]RHL part is coming as zero but center part is coming as two. So is it going to be continuous? No. The answer is it is going to be discontinuous. So here we can write this.

[00:41:16]Then how to represent anti?

[00:41:18]Representation is limit x ts to 0 - fx is equal to limit x ts to 0 + f of x but it is not equal to limit x ts to 0 f of x story over here only understood so center part is not satisfying l is satisfying equal to rl but center is telling no no I'm not continuous got it so like this also you have to solve.

[00:41:48]Come on, go for it. Please solve now.

[00:42:00]Did you got all continuity questions selected like this? Are we ready for next pattern questions? Get ready, please.

[00:42:20]for you people. One part question is checking continuity. Another part question is find the value of the unknown variable where it is continuous.

[00:42:30]So these are the two topics you have to discuss. Right now we are very busy in checking continuity. Correct? Signal function continuous or discontinuous?

[00:42:46]Critical point we are checking critical point is discontinuous. Now let's talk about another question please. Uh this is a very beautiful question for fools.

[00:42:56]They have given ready for this searching one very idiotic question. I'm not getting that question.

[00:43:07]So this is NCRT exampler question and this question is one of the cutest question of mankind. So now look here cutest question f of x = x by sin 1 by x. You have to check continuity when x is not = 0 and 0 when x =0. Check continuity at x =0. Come on.

[00:43:36]Very cute question.

[00:43:48]Zoom out.

[00:43:55]Zoom out means big or small.

[00:44:16]Back to 110%.

[00:44:23]Come on, solve this.

[00:44:31]Solve this expectation without solving them. People will say it is discontinuous but answer is not it is discontinuous.

[00:45:07]Answer is opposite of discontinuous.

[00:45:11]But how it is going to be discontinuous?

[00:45:24]She's telling I'm feeling sleep. So that's what I'm saying. If this is not for sleep, this is for coma question. In board exam, if this type question coming means you will not go in sleep, you will direct going hospital after result will come.

[00:45:41]Oh, you're in 11 continuous. Very good question. In life, everything can be poisonous. More study is the only thing which is never poisonous.

[00:45:58]Shall we solve this question please? Now look here nothing big we have to do.

[00:46:02]Again our story starts with this part LHLC. So here again we are going to draw the line. What is there that at critical x=0 value is coming as zero. So here x is less than 0. x is less than 0 because here it is sigma sorry sine function. So f of x = x sin of 1x. Same left side also and right side also f ofx = x sin of 1x. Okay. So left side and right side same function you are going to take it is not modus. So plus minus will not come. Same function will come.

[00:46:44]Huh. So now look here we can solve LHL LHL at x = 0 what we will write limit x ts to 0 - fx and that time we are writing limit x ts to 0 - f of x means x sin of 1x done left side same function right side same function understood center part is x =0. So now let's check this one out.

[00:47:20]What will we write? Let x = 0 - h since x ts to 0 minus. Therefore h ts to 0 for all h is greater= 0. So here we will write limit h ts to 0. Now here what we going to write? limit h ts to 0. Here it will become 0 - h sin of 1 by 0 - h correct. So here it will become limit h t - h sin of 1 by - h. Now we know that sin of minus theta is minus sin theta.

[00:47:57]So minus will come out. It will become limit h ts to 0 h by sin of 1x hy because sin of minus theta is minus sin theta and minus and minus will become plus here. So till here it is okay.

[00:48:13]After that what is the story? You need to understand one thing like whenever you talk about sign. S is a continuous function. S is a continuous function.

[00:48:24]And you need to understand that here h is not written. It is 1 by h. So 1 by 0 mean the value which is not going to be coming for any specific angle. Correct?

[00:48:35]Now it is coming till infinity. Direct infinity. Infinity also sign will be continuous. In infinity also if I can draw like this I'm drawing like this.

[00:48:45]I'll keep on drawing like this. Still where I will be I'll keep on drawing this only. Now why I should be changing?

[00:48:51]So if any angle so what is this one?

[00:48:53]Whenever you write sin theta sin theta means here this is specific angle.

[00:48:58]Understood? So here the angle value is going infinity in infinity also you will stand and say that hey the value will lie between minus1 to one. That means the angle will oscillate between minus1 to see whatever will be the angle value sign value will oscillate between minus1 to one mean minima to maximum what is the minimum point minus one what is the maximum point one so what with what curiosity we are going to write the answer we will be writing only one thing by this angle is the biggest angle this angle cannot be handled one by h angle cannot be handled it's a very infinite angle infinite angle angle also means what we will write X value will come as 0 0 into oscillation between -1 to + one correct now because we cannot specifically say oscillation of number between what -1 to + one so oscillation of number between minus1 to + one only will come now no comment became Now I hope everybody understood oscillation of number between oscillation of number between minus1 to + one correct now whatever number will come it will come between minus1 to + one understood and that number into zero because h is there now so that number into 0 will become zero understood center value already given zero so this side also coming zero. This side also coming zero. RHL we are going to check RHL at X =0 limit X t + f that will be limit x ts to 0 + x sin of 1x. Again same story love story we are going to repeat. What is our love story here? What we are going to write?

[00:51:01]Let x = 0 + h. Since x ts to 0 + therefore s ts to 0 x ts to 0 + therefore h ts to0 for all h is greater= 0 then here it will become limit h ts to0 h by sin of 1 by h then what you will write 0 into oscillation of number between -1 to + one again answer is coming as zero. So left side center right side with very much dedication the answer is coming as zero. So it is continuous. So last line what write LHL at x =0 is = rl at x =0. So it is continuous. Ll at x=0 is equal to rhl at x=0. So it is going to be continuous. No other thought. Now correct diagnosis.

[00:52:06]Cancer is detected and it is continuous.

[00:52:33]Your brother is saying uh push is telling or brother is telling if possible keep the class at 11.

[00:52:44]Your sister is having class at 11. Your brother is telling take class at 11.

[00:52:49]What a type of mystery for me.

[00:52:52]Today I'm going so I have to take class at 10 10 mostly.

[00:53:04]Ready?

[00:53:06]This one doesn't have okay. Can I remove this?

[00:53:14]Write a question.

[00:53:19]You check is it continuous or discontinuous?

[00:53:23]Question is f ofx= e to the power 1x. Such a beautiful question you know most beautiful question more than any flower. This question is beautiful.

[00:53:43]When x is not equal to zero zero when x= to z come on start solving if this question is coming mean beautiful butterflies will start running in your stomach.

[00:54:02]See such a beautiful question I have given.

[00:54:08]See when I was in your class and all now only these type of things like wow what a beauty what a beautiful question like this only I was thinking oh such a be beautiful question now solve Should all the mothers write this song?

[00:55:06]Can I may I solve this question?

[00:55:10]Okay, I told you only one thing from the beginning I'm having this habit and believe me from graduation to everything I always got success when I'm solving continuity means one extra line writing is never poisonous but this extra line will save you from all troubles It's continuous.

[00:55:42]It's continuous because discontinuous continuous okay that's why I'm telling you don't that's why I'm telling you don't only keep zero zero continuity left side right side also sometime it is very poisonous and for that proving only I gave this question so look Here I'm starting the question LHL at x =0 or it will become LHL at x=0 limit x ts to 0 - f of x which will be written as limit x t - e the power 1x -1 by e to the power 1x + 1 done till here after that what we are going to do come This is a very beautiful question again and again I'm telling all the beauty will go if this fool will come understood or not if this full question will come means all the beauty will go that is x=0 minus h since x ts to 0 minus therefore h ts to0 for all h is greater= z real mathematics is here only let's check this one out now look at this still we done now so that means it is going to replace limit H ts to 0 came from here only see everybody please Ruda did you got it what it will become e to the power 1 by 0 - h -1 by e to the power 1 by 0 + h + 1 done now with so much love and affection with so much love and affection just understand the power - 1 by h 1x h E to the power 1x h is very dangerous because it is coming e to the power 1 by 0 e to the power infinity understood e to the power infinity is infinity only got it but but one thing is there if if I will write 1 by e to the power 1x h then it will become 1 by infinity and 1 by infinity will become zero.

[00:58:18]Okay, she wrote discontinuous continuous still. So Kushi versus Kashish. Kashish is telling continuous Kushi is telling discontinuous. Now any one person will win here today. Okay. So let's see. I told you one very important point one by infinity see this is the most important core concept here important I'll discuss so here I'm going to take first I'm taking out 1 by h e to the power 1x h I'm taking out so e to the power - 1x by h I have taken out so it will become 1 - 1 by e to the power - 1x h down also e to the power - 1x h taken out 1x 1 + e to the power 1x h so this and this will get cancelled now here it is coming limit h ts to 0.

[00:59:36]Check nicely. If it ts to 0, it will come 1 - 0 by 1 + 0. Is it correct? 1 - 0 by 1 + 0.

[01:00:06]Hey ma if I will do like this mean it is trouble correct only it is going to be trouble it will come up it will come this 1 - e to the power 1 by h and 1 + e the power this part again minus will come 1 h it is more dangerous we can't take anything common here only story E the power the equator e the power - 1 by h -1 e the power -1 by h + 1 is why we need to take common directly we can take reciprocal for that only I give concept what concept I gave I told e to the power - 1 by h is there and if h is tending to zero that means it will become e to the power 1 by 0. So it will become e to the power infinity and e to the power infinity is infinity. So very easily we will do one by we will write here 1 by e to the power 1x hus 1 by e to the power 1x h + 1 that's it because negative was there now so when negative is there I will shift it reciprocal it will become positive so now we will do that one next step is 1 by e to the power 1 by 0 - 1 1 by e power 1 by 0 + 1 + 1.

[01:01:49]So here what it will become 1 by infinity minus one 1 by infinity + 1 1 by infinity 0 - 1 by 0 + 1 answer is min - 1 see for you I have written good morning for you I have written every steps every terms and condition now is it going why I will take one by h common yeah this I why I for what for entertainment no if this situation is there minus infinity part will come here minus infinity always it will again ts to infinity only but zone is minus infinity understood or not so here the first important work is whatever in the head negative is there convert it into positive understood so take reciprocal after taking reciprocal substitute the value of h and then you are getting it minus one so now rhl at x =0 So limit x ts to 0 + f of x and that will become limit x ts to 0 + e to the power 1x -1 e power 1x + 1 let x = 0 + h since x ts to 0 + therefore h ts to 0 for all h is greater= 0 so same story I every time we are writing this so I hope you people have understood this one so now till here is not at problem. Now you see here limit x ts to 0 e to the power 1x -1 e to the power 1x + 1 by infinity - 1 by infinity + 1 is coming correct. So this time we have to take something common. What we have to take common here?

[01:03:41]Now take e to the power 1x h common.

[01:03:45]What it will become 1 - 1 e power 1 by h 1 + 1x e power 1x h these two cancel now beta if this part is coming infinity again 1 by infinity because 1 by 0 will come now see here from there part I'm just writing this section very clear clearly see understand this part till here I hope everybody will do so limit now limit is going it will become 1 minus 1x e power 1 by 0 1 + 1 by e to the power 1 by 0.

[01:04:21]So what it will become 1 - 1 by 0 means what? It will become infinity. It will become 1 by infinity. And here 1 + 1 by infinity and 1 by infinity is 0. So 1 - 0 by 1 + 0 answer is coming as 1. And clearly if you see left derivative and right derivative so left limit and right limit our left limit value is what? Our left limit value is minus1. Check left limit value is minus1 and right limit value is 1. So LHL at x =0 is not equal to rhl at x =0. Therefore f of x is not continuous at x=0. I rest my case. So only I told you this is a very beautiful question.

[01:05:16]Beautiful question is beauty will go only foolishness will come. So you have to understand modulus is a different story. Exponential is a different story.

[01:05:26]Exponential to the power zero is one but exponential to the power infinity is infinity. Are you understanding?

[01:05:34]Exponential to the power infinity is infinity. So we have to keep do one thing in that case. We have to convert it into one by exponential. One by exponential and one by exponential will become what? Zero. So that's the technique we have to follow. I hope till here everything is okay. But I'll be conducting online class. There's no holiday for you people.

[01:05:59]2 days I will be trained means wherever I will be if journey means okay the moment I will be landing to a safe place meanness I will conduct your class okay 10th gave me threatening don't call us for 7 days but 11th and 12th no bargaining okay that's it for today bye-bye everyone