JEE Maths | Continuity L-1 | Introduction | Chapter 5 Maths Class 12 | CBSE/JEE Main 2022 | Vedantu

How are all the children Write it once for sure We are beginning your calculus class Why Because if you are targeting it for 12th CBSE Yeah So you know about your 80 marks final paper 40 marks are calculus 40 marks when said 5 chapters are calculus If you are targeting JEE you still know Even this year the JEE January 2020 main that took place So it included your AOD your integral calculus differential calculus which had the highest weightage And of course for NDA as well it follows the same pattern It occurs a lot of time it has a lot of weight This section is the most important It should be even higher than the highest enthusiasm So we are moving this entire class to 5 pm Everybody are we excited for this kids Let's start again But before starting we would like to tell you What is the value of one rupee Think about it buddy you'reading something And while reading you might have gotten stuck somewhere Feel like no buddy this is my portion I definitely need to have this a bit clearer I keep reading and reading I thought this part was in Conics man I don't understand it at all What if I get what if I can get access Of that particular video for just one rupee do you think it's possible Well yes it is possible when you are here Then you are under this video So here is the video you are seeing on this page Below here when you go you will get the link to this micro course Here you select your class As I have selected grade 12 for the sample Let's try physics today Reading about electrostatics Reading about magnetism and matter The magnetic effect of current is to be read One rupee filled with attention But if you want you can select it here What Vardant has done for you Check it for sure But before talking about him come here Let's watch a little movie Because watching movies is a lot of fun for us Who is this movie's clip that we saw Take a moment to pay attention here Now I will tell you why it is shown to you Now what's happening here kids What does he wantمحك to see who But the brakes keep failing As it happens with you people But Boffring Azea what happened Hey your flow has broken down Your flow has broken in the middle When Flow is selling it is a complete test Is this not continuous Does this make sense to us The meaning was continuous Continio T is basically a When you are continuously flowing without any bumps without any jumps do we understand Absolutely without any break Is it clear Then today we are going to study about the continuity of a function at a point at an interval Then we are going to build upon this topic And then we will have many questions based on it Are we excited Yes Everybody are we excited Let's get started here Heads up Understand the first point Then move on to the interval Right Understand this one thing here We have understood this much Then I will also give some examples To study continuity How many children are here Tell me once again about the jar in the chat box I wonder what the children are writing here Mayor Lee is writing yes here If you need to revise it at all Wherever there is a need it will continue to be there At a certain po Left side from Here so far Here as well You will see it here There is a short break So keep it continuous You know both of them should be equal I know that for sure That is the value exactly at A That small gap we had in this area It's also filled up The point is if all three match So it means that they were approaching from the left side and the right side So this is your third missing concept That means if you look at it by splitting it up When should I use this definition When do I need to see the split It became clear to everyone that it was a big deal This makes sense to us Such a basic thing When we are developing a new concep Did everyone understand that So we call this a We will use this repeatedly in questions So remember this This thing was understood by everyone If I need to split this further A method with an h also exists for the left-hand limit You have done it even within limits Where h tends to 0 and you write this friend And for the right side暮 What can you do for R H L It means the same thing that a is a bit bigger that means on the right side smaller than a meaning on the left side limit tending to a f of a plus h Then where it will be needed we will use this definition also and value of the function at a These three match up so we get continuous now usually you know this is direct It leaves us this directly leaves the work In fact by combining these two We can write this as well Please make a note of this Everyone and everything Understand slowly Now look here buddy What's happening next let's try to look at the graphs that are here Here what you have written is equal to l and the value of the function is also l It's clearly smooth boss this is continuous at point A Now look at this one why is it discontinuous Because there is a gap here Can we see point A On the left side and right side they were approaching the same point But at A when I entered the value I did not get the same answer so there is a gap ere is a hole in the line in the graph so it is discontinuous in the same way children here The left side and right side limits are different here can you see this Here the limit on the left side and the limit on the right side are very different Look they are different Answers are coming so here is the list Functions must be discontinuous as well correct I am just trying to visualize through different kinds of graphs here Now let's move a little further ahead what are the reasons for discontinuity Because often they ask us in the paper Is it discontinuous or not and at times what kind of discontinuity is it so understand A function can be discontinuous There are many ways Is it How many ways can it be discontinuous It may be that the limit is not existing It may be that lhl and rhl are existing but not equal to the value of the function at a there can be many reasons so one of the reasons that you see is here look kids once Just look at this graph because you will be able to follow this one here now At some point look here there is a discontinuity here as well Here are different types here as well What is it What's going on here kids LHL and RHL are not the same Look at this on the right side The graph is coming in and it's trending upwards here Those from the left side of the site The graph is rising it is trending upwards here So at point A LHL is not equal to RHL there is no discontinuity correct in the same way I will check the B now That's all correct on B Look at the left side as well The match is on the right side too The match is but the value of the function at B dear this is f of B No there is a hollow circle here that means the f of B is not defined So there's probably a discontinuity here I'm just telling you I am showing examples of how there is a discontinuity here It might be right now if I 神奇 go ahead here look at C Look at the root on C What's happening on C LHL RHL Then it is uncoverable on C but f of C is also defined here Look if I move at this left side I come from there let's say you take this Approach from the left side with C value 2 right Approach from the site even then value 2 but value of the functio At C he got out and buddy he came in 3 in this case It is also a discontinuous function In short we need to keep that thing in mind that LHL RHL and the value of the function A should coincide the point is a continuous one The graph should come right Now a little further We are moving let's talk about types of discontinuity As here removable and non-removable It means you can delete someone You can't remove it buddy here We won't go into the details I'm just telling you a little bit and then we are coming to it e questions check so removable discontinuity when do we say well limit thenighted It exists but it is not equal to f of a that means the left-hand limit is equal to the right-hand limit but the value is not equal to e value of the function at a okay and where you look you where you look isolated point discontinuity That is the point where your limit exists where even you are doing but they are not the same they are not the same and here is the missing point tha f of a exists but f of a does not exist friend here f of a's There is no value yes f of a does not have a value and non-removable discontinuity at If I come okay this is slightly detailed but this is After that you have questions right so discontinuity of first kind means when LHL and RHL exist but they are not equal to b correct they are not equal That is understand it this way That LHL at a s not equal to RHL at a this thing became clear to everyone LHL at a is not equal to RHL at a we call this a junk discontinuity also You can't remove this everybody yes everyone eternal son this session is concept builder So we will start from the basics and then come to the questions okay everyone This thing became clear and the second kind is That does not exist for example Look here at tan x at x equal to pi over 2 e value of the function at pi by 2 is not defined so this is something which is non removable this Discontinuity remains like this take the same way look at 1 over x squared Look here the values are approaching but they are not exactly the same and at 0 as well Again the function is not defined So these are few examples When you don't have continuity correct I see let's Let's start with the questions now you ask so many questions and let's see how We target them guys let's do this Let's understand this Okay so let's understand this it says check its continuity at 0 and 1 which means is my function continuous at 0 Check for the smallest value at 0 yes at x equal to 0 let's Check this at x equal to 0 so what will be your left-hand limit at 0 LHL what is going to happen to me this this functio limit x tending to 0 x squared plus 2 because The left side of 0 is coming function this is going to be what 2 RHL at 0 low because there is a spl In the function we have to look at it separately RHL is above 0 means x is greater than 0 so this is going to be obvious man this will give me 3 Tell me here in the chat box what is the value At zero write in the chat box let me see how many of you can understand what is the The value of the function at 0 Exactly now there are many childre The answer has come absolutely 3 absolutely It's 3 isn't it looks like Because here x is given as equal to 0 Just like that It had become clear that the limit did not exist Now it's definitely clear from here so the function is discontinuous at x equal to 0 right now children let's talk of x equal to 1 same way you LHL And RHL look at it yes LHL at x equal to 1 what is What is this LHL at 1 3 is showing up here RHL at 1 It is going to be e raised to a power if I take the limit if I do Yes s is going to be 3 times 10 to the power 1 so 3e-2 absolutely They are not playing at all They are not matching and of course your f of 1 is also the same at means the function is discontinuous at 1 also yes everyone this thing made sense the function is discontinuous a x equals 1 also Let' s move on to the next topic So this is what we have discussed Yes let's continue check This is your function check if it's continuous at 1 and 2 and if there is any discontinuity then find the type Let's see let's see how many children do this Yes start with 1 Let's start with 1 yes what will we do at x equal to 1 that is if I take LHL at x equal to 1 what will happe He will come from here limit extending to 1 tangent of pi x by 2 man this is jus an pi by 2 will be undefined which is not defined It had already been changed There is no limit is there Take a look at RHL as well you know There is no point in taking it because we have got this to be not defined Here keenly as well There is no need to check the value of the function it is discontinuous now it is discontinuous then which one is What type of discontinuity is this or is it non-removable why Because my LHL and RHL Meaning LHL does not even exist Correct now let's check for x equal to 2 Once tell me in the chat box is it continuous or discontinuous at x equal to 2 let me check tha Yes absolutely it's the simple questions that make the fun ones Questions will be asked so what will I get over here what will I get over here this It is going to be LHL if I see what is happening here limit x tending o 2 x minus 1 then it will become 1 and for RHL limit x approaching 2 1 over 2 minus x is undefined the same problem aga It's discontinuous and what is a non-removable discontinuity Yes why because limits do not exist If there is no limit then tell me from your side How will you remove him yes absolutely then this is what we have Sirf basically was to write here also undefined essential discontinuity which is a kind of your non removable discontinuity at that point let 's go ahead and expand this in an interval What is the interval man When do you call a function continuous on the entire interval we had just talked about the point okay Then of course it's clear from the graph what else do we ow AB if here interval Hey I'mılı here I didn't do anything in the middle For the point C belonging to AB anything random and I get to know that my function is continuous at C Yes now I took a random point This means it is continuous at the interval at the entire interval okay Now I ask you a question look at this example And tell me if it is continuous or discontinuous in the interval AB by looking at the graph Tell me if it is continuous or discontinuous in the open interval AB let me check how many children answer correctly Look here look at this It's a lot of fun very much so All the children are also discontinuous You are writing yes a lot of discontinuous also Why because you have done this The graph shows a break yes but but my question is about the children I asked you about the open interval AB or open erval AB means that the endpoints are not included an open interval AB means A and B and B they do not belong to AB if you leave the endpoints yeah the rest of the place is my functio It is continuous it is a continuous function so yes it is continuous so I am give Check in the interval Something is happening outside of it I don't care yes if my B had been included my B yeah man this is discontinuous because here I have to jump It appears break it appears in the function but this is of course a continuous function correct This thing became clear to everyone because we don't include B right in the same way So basically what we said on the right side is continuous at A and on the left side at B overall This function will be continuous on the closed interval if from the left side of A and from the right side of B as well It is continuous when it is on otherwise it is also 100 in open Correct let's go So before I go ahead children once again a quick reminder if you are for JEE 2020 You are appearing then we have something which is which I call as A gold mine why because dear Your revision is very It is necessary your level of preparation Anything you a lot of preparation your awesome The fund has been running for the past two years This time your preparation is ready friend now is the time o actually revise everything and revising in an efficient way not that something is coming from here today Revised I took the book that was here yesterday That's all random yes So to actually find a method o this madness to this randomness for you or this entire thing We have designed a course such that when you enroll you will get the best experience Well look I've told you many times You get the entire timetable Now the next 40-45 You get the entire timetable as you have days left You are also watching here The class is ongoing so it's like when you enroll You get a basic reminder on how to attend the class Your entire class schedule is set if you have already f I want to watch kids just a second if I want to watch like There's an old class I go to my sessions here There's an old class man What you want to see again is from you It has been missed for instance the class that took place on June 2nd you can always replay this yes replay the video and after the video same here Your notes are also uploaded Which thing you have a teacher He had taught exactly the same way children yes Exactly the same way your revision was okay so absolutely everything s shared with you plus assignments you receive You get a test that's it Powerful questions that you have already asked in physics chemistry and math Then you're all set right Then buddy this is the chance Look because the course just started if you are planning to revise then this is the best way you can right so What can you do you can do this Go to the bottom of the video you get this link vdnt in/jeeccе It's a link to a crash course You click on this you click on buy now it's there after that You can run my code nagcc which means Neha Agarwal Crash course and you can get all physics chemistry math at this price okay so this is like "Just think about it" Then what I keep saying every time you are 40-45 Invest 100 rupees for days your best For revision because here you have ia They are reading themselves right Let's talk about problems now whose solutions We need to find it okay this is the next question Let's find this Exactly kids it's still the crash course that just started It has happened so this is also the time that you should take one or two The backlog of videos you have during the day is one that you can replay You can join right away so this is actually the best time okay so this is what is given to me this is already continuous mush Then you have to find the value for it I guess many children have found it The 4 by 3 answer has been found absolutely First of all if the function is continuous that means the limit approaching 3 from the negative side the left side limit approaching 3 e value of the function at 3 is the same as the RHS so what happened here Look on the left side what is coming this is your x2-1 and limit extending to 3 from the right side is 2ax and value of the function is also same just the same as this Just do what you need to do find it Yes replace it with 3 so this will be 9-1 that's an 8 and here you have 6 from so a is 4 by 3 that's The answer is done yes or no Save the people everybody Tell me once and make sure you are hitting the like button if you want more such videos look we are planning to change your time to It's 5 pm so if you want me to be very regular with these sessions the You also need to arrive here regularly correct so 4 By 3 answer let's go next question on your screens now check the continuity of the function yes Rohit of course you can join the session so if I am given this check the continuity of the function the first thing is to save where will I check it's there where my function is splitting Will I check it my function basically here 0 is the only point Your answer is correct so if I take the limit as x approaches 0 okay e raised to the power of 1 over x minus 1 e raised to the power of 1 over x plus 1 I am following the rules x is tending to 0 for now okay So what came here buddy if I take e raised to power 1 over x common I will have 1 minus e raised o the power 1 by x is equal to e raised to the power 1 by x common here taken as 1 plus Why did all this happen and here's the value You are coming from the positive side now Okay so this What will happen look at this This has become infinity repeal It also became infinite here They both canceled out I got answer one My answer is ready now if I am here Let him go look here the value of the function at 0 It is zero so it has already been shown that they are not equal Yes it was already known They are not equal Absolutely then now you Whether you want to take LHL or not we know that this function is no continuous at 0 is not continuous only at 0 okay Everyone understood the matter By the way if you take LHL then it will be minus one You can make a reservation here Meaning if I am going from zero minus all right let's go forward here is the next question let's see yes DP you ca Let's go kids How are we going to solve this now ow one plus mod sin x and raise to power a mod sin x here I need to check zero o Please submit your application Then I will read you LHL and RHL Lahla Zero You are given the gift of wisdom and it is continuous This means zero or lim Extending to 0 if I just go with that mod sin x okay Now I don't know how many of you kids get this trick Because if I set x to 0 it becomes 1 raised to the power of infinity What is the trick with 1 raised to the power of infinity When a function is raised to the power of another function So I call it the tree and monkey trick right We call this the tree and monkey trick meaning that if your functio raised to the power of another function 1 raised to the power of infinity in the form of So what are you doing The answer is E Raised to power limit whatever your limit is like here is 0 You will call this monkey down from the tree it is climbing It will hurt you yes so you cut this tree So this tree you cut down it's minus 1 and this monkey comes down Yes this monkey comes down absolutely this is the way to remember So what happened here this becomes E raised to the power limit extending to 0 This is my monkey dude who came down and this is my tree Although we should not cut trees it's just a way of remembering 1 plus mod sin x minus 1 because tree minus 1 is you know both of these have canceled out Here I can see that both have been canceled So e raised to the power of a is the answer the first answer has already come to me E raised to the power A from LHL at 0 everyone understood it so clearly Yes everyone you see you let's check out RHL at 0 right and eventually we have to equate everything because the function is continuous What will happen to RHL at 0 limit extending to 0 E raised to the power of tan 4x and here is your tan 6x or look if you divide by x Yes if you divide by x and multiply by 4 from here Divide by 6 then you'll know if the limit extends to 0 Then it will be 1 here it will be 1 here Essentially I'll be left with e raised to the power of 4x6 which is e raised to the power of 2x3 If I want it to be like this you will put a limit monkey trick here Then E raised to the power limit will come won't it between them They will be canceled anyway directly E raised to the power of 2x3 is coming Now what are we going to do guys it's finally right B is equal to E raised to the power of 2x3 which is equal to E raised to the power of A So it seems clear here E is equal to 2 times 3 right and what will B become B will be what how will you find B it will become 2x3 if I take Yes everyone sorry B what will happen to you here Type "Jada Likho" in the chat box everyone B is E raised to the power of 2x3 that was straightforward all along wasn't it So basically what do I have here kids On solving A is 2x3 and B is equal to E raised to the power of 2x3 Then they will move slowly go forward and in the upcoming sessions we are going to keep raising our curve But before I go on make sure you've subscribed to the channel and you have hit the bell icon yes and at the same time if you want more and more such videos Then it's up to you boss let's go Sixth question now everyone look at this Jada What's going to happen here just watch How will you do it Hey if I come to 0 minus Actually minus does not come with log What's going on with you If sign log 0 is occurring log 0 is not defined This is not defined boss it is clearly discontinuous Just say it straight out this function is discontinuous Yes or no in the chat box kids this was easy If we know that log 0 is undefined then absolutely there is nothing You have 0 minus 1 0 plus 1 the same thing is happening it is not defined correct let's go to the next question here it is come on let's see this Let's check this one Yes let's check this one now Look fx is equal to 3 raised to the power of x minus 1 all squared sin x log of 1 plus x is continuous at x equal to 0 So what is the value of the function at 0 is it so that's the question so hey continuous is the meaning is limit extending to 0 f of x will be equal to value of the function at 0 Yes or no save it Everyone understood the thing that means Now look here Now there are many formulas being applied here if you see limit extending to 1 your 3 a raised to power I provide formulas to be used limit extending to a sorry limit extending to 0 a raised to power x-1 upon x what happens here This is your log in right same way for children if I have limit extending to 0 sin x over x This is the logarithm of 1 plus x divided by x with the limit extending to 0 This is your 1 Everyone understands this exactly Here is the combination of all three so what should I do It also needs x x squared is needed because the power is 2 So that's what they do 3 raised to the power of x-1 squared on x So what I am doing is I am dividing my numerator and denominator by x squared That is very correct Because here sin x will take an x and log of 1 plus x divided by x this will take an x yes Everyone understood this thing meaning the children These are your results log 3 squared this is also 1 Then that means my answer is this Please don't make this mistake now That guy brought this 2 here And said 2 log 3 there is a very high chance When a mistake is made in this question That children speak Good people are in power So we write 2 log 3 watch my video logarithms in one shot It includes all these properties I have discussed that I have mentioned this carefully that children log x raised to the power n The logarithm of x is n But if you raise log x to the power n nothing happens It remains as power power only comes down when it is not with log but with the function inside These things make sense everybody yes I hope we understand this okay let's go forward all right This is the next question on your screens now Let's check this out kids Let's check this out hey e value of f of 0 so that this function is continuous at each point on its domain I think this also affects many childre It will become clear why if you have this property It reaches its limits again dear I need x Then the series is continuous so I can directly say it's the same as f of 0 Yes f of 0 so limit x tending to 0 if I divide numerator and denominator with x en what will happen 2-sin inverse of x divided by x the whole divided by 2 plus tan inverse of x upon x completely like your sin x over x if the limit x tends to 0 or tan x by x if limit x tends to 0 same way even for the inverse function the property holds oh come on This is also 1 this is also 1 so I'll directly get 1 by 3 which is again B for bingo please make a note everyone Yes everyone this thing is clear to all let's go Let's go ahead and see what are we going to do now all right One minute okay this is the next question on our screens now let 's check this one out How do wezia do it it is continuous to find the values of a and b and f of 0 Ayuash we have already held the sessions in the morning So at 11 am and your 11th or 12th session we are shifting them in the evening Okay everyone J at 11 am every day okay what are we going to do here look carefully at this it is continuous at x equal to 0 To find the value for f of 0 it means one thing became clear If it's continuous children then this limit will exist and it will be equal to f of 0 so the limit will be certain okay and will give us a finite value then if I substitute Yes if I set x to 0 So this is 0 plus a into 0 0 plus b into cos 0 is 1 upon 0 or this is so There's a mistake if my function is finite yes if my function has to be f it is finite then my b must also be 0 Then only will the 0 by 0 form arise and that means we move forward You can apply the LH rule We can do something else ahead But if this is divided by zero then it is undefined this is the way It will shut down so the most important thing is that the value for b has to be 0 This thing is clear_HORIZONTAL to everyone yes b is directly 0 You see what is my function I set b to 0 this entire term has vanished you see limit extending to 0 sin to x plus a sin x upon x cube if you wa The LH rule now comes into effect in 12 Allo will not be given by CBSE ISC allows allo CBSE I don't do it the I'll also tell you another method for him But if your LH rule is allowed it means this is a 0 by 0 form The LH rule means it's above and below Differentiating if it is 0 by 0 or infinity by infinity Exactly what children say The LH rule is our best friend isn't it what did you do Differentiated 2 cos 2x plus a cos x upon 3x squared limit extending to 0 Now I am stuck aga I will enter the value to check Then look at this cos 0 Again 1 this 2 will come plus cos 0 again 1 so 2 plus a upon 0 This is a mess this is It will not be defined again oh this is a problem isn't it the meaning of this e entire thing should be zero my a and b are sorted yes everyone understood this a and b have been sorted ow what child do I have what is my role children yes What is my function Now look here B has already gone a is also yours minus 2 has already been that means sin 2x minus 2 sin x upon If you cube it directly find it Then you can find f of 0 right that is if I directly show that to you These are the ones who have arrived Then my function will just be limited to this my function is only limited to this I'll simplify it and I will get my answer as minus 1 Right yes or no in the chat box Okay some children are talking I'll tell you another method If you are taking the 12th CBS this method will not be allowed It will go that far with you Up to where b equals zero you are absolutely fine It should be 0 by 0 anyway To proceed you cannot simply apply the LH rule So that's your zero buddy you have sin 2x plus a sin x x cubed has arrived you can split sin 2x 2 sin x cos x is look here Here is the alternate method for you you can write this as 2 sin x cos x plus sine x from here so this method is for 12th CBSE This method is for 12th CBSE right what will you do now I can take sin x common What will be left inside 2 cos x plus a This sin x by x has become 1 sin x by x is 1 This is 2 cos x plus a over x squared if I plug in the value again 0 over 0 Then it will give you that a plus 2 equals 0 so a is minus 2 There is just one step different for you all that this method It will have to be done It's entirely the same this thing is understood by everyone it's entirely the same Let's note down the homework question from the notebook Make sure to comment the correct answer in the comment section below and if you want this continuity to continue Don't break you also have to be regular and make sure you are here at 5 pm this is our telegram group Where my notes are shared If you are a master then what can you do First download the Telegram app here you will find this link Clicking here you can join the official group of Vidan Tu Math Some children say It is very important because your Mobile phones are also very useful tools When you join Grub Then they come to a section by linking Click on the links You will get all the links the ones we share Make sure you are downloading the learning app it is very important because you are your own You can also make a mobile phone a very useful tool with very effective apps Your live quizzing has already started here you get a lot of study material a lot of free live sessions That work has to be done And you can trust Google Pay very well Because Google gave us the User's Choice Award You have to make sure Save yourself make sure you join And the JEE timing has been set Suva at 11 o'clock every day and your 11th and 12th will be at 5 pm This task must be done That the children are having fun and they want more such sessions So make sure to like that button Hit the like button as well subscribe and I will see you guys super duper soon till then thank you so much Babaa