Hello hello hello students how's the voice is it okay screen will come is the voice ok students voice and screen both okay hi ravindra okay basha and rohan is there hi pavitra hi gokul.
Okay so sound and the screen is okay so we are about to enter the calculus part so let's see okay so everybody ready where is nishat not there vimal is there lucky is there surface timing.
Definitely definitely and those timings will be fixed and for students the timings will be fixed google meet links will be given tk will fix driver yeah okay everyone ready what happened.
One second just so let's start the chapter limits very simple very easy first let's understand what is a limiting situation okay and what when do we say limit exist okay so the first thing that comes in mind what is limit and why do we need a limit a a very basic introduction will go okay so what.
Happened is i'll show you already everybody cello cello cello ready one second one second just let's see everybody let's start let's start let's start all good everybody shalom.
So first thing let's understand the limit story so what happened is a lot of times we got functions where at certain point it was not defined at certain point there was a gap function was not defined now we wanted to know the mathematician they wanted to know.
At this point where the function is approaching okay i don't know exactly what is the value of f of 2 i don't know about that where it is okay it is a question mark for me but can i know limit x tends to 2 fx means the limiting value if i put to this function means.
How far it is okay the value at 2 if i want to go to this part or understand like this let's say this is let's make a map this is a house here and this is your house here okay you don't know exactly this is your friend's house.
And this is your house so you know nearby your friend one lake is there small lake okay you don't know exactly your friends house so what do you do you go keep asking people where it is where it is where it is where the lake is okay you can say oh lake is there now your friend.
Has given certain idea that from lake from lake it is nothing but just five minute walking distance so now you can have an approximation that lake from my place is somewhere around let's say two kilometer and from his place from the lake to his place is like a five minute distance and.
If i'm walking at a speed of what you say uh one kilometer per five minutes or like 500 meter per five minute so it is like 2.5 kilometer from my place so you got an approx idea how far it is so in the functions also people wanted.
To know exactly if the function is not defined it is breaking at 2 let's say it is breaking at 2 but what is the function approaching here to know that approaching value okay they started the the concepts of limit so basically the concept of limit is to.
Get the approaching value i can't get the exact value maybe due to any reason there's a grape now what is a break in the graph there's something else whatever it is but i need to know if there wouldn't have been a break what should be the approach value clear everyone now we'll see.
When a function is when a limit exists and all that okay so basic condition was for functions we wanted to know whenever there was a break in the functions and at certain points function was not defined so we wanted to know what is the limiting value or approaching value to this function another example sir okay let's see.
Another example take this example okay this is your value 2. now what happened is no issues that's here so what happened is this is you don't know.
Exactly okay you don't know exactly what is the function value at 2 you don't know about that because the function is telling that the function value at 2 doesn't does not exist for whatsoever be the reason we don't know about that but can we find out the value which is just near.
To two you'll say sir okay function value at two is not known what about function value at one point nine nine nine nine nine nine very near to two if i can get that okay i said okay if i put it in this function almost it is ranging approaching to almost it is approaching to 4.
You said okay so this value a bit smaller value than 2 it is approaching to 4. what about the bigger values just a slight bigger than 2. you said okay so what about 2.0001 can we check on that okay we did a checking on that we got it exactly this time also almost 4 almost 4 so can we say.
That if limiting value if i am finding out there is no break or something if it is not there then this value will be almost tending to 4 exactly f 2 is not 4 but the limiting value because from the left hand side if i'm approaching the function or from the right hand side i'm approaching.
Both way i'm getting this value 4 only clear both way i'm getting the value for only so i was able to get an idea what the function value can be or from where it can range to clear students so basically limit is to get an idea about the approaching value okay we can't get the idea about the exact value.
But we can get one thing a bit before a bit after and then we can predict okay a bit before is almost reaching to four a bit after it's almost reaching to four so the limiting value will be around that region clear got it everybody what about others yes no yes no synthesis of 11 topics so don't hesitate.
To ask doubt okay we'll go these all we won't need but still understanding is important okay so basically limit we started only because of this reason functions where a lot of times at what is a lot of places they were not defined so we wanted to know where it is defined a bit before and a bit after if i get to know i can predict.
Okay it becomes it becomes easier on our part clear okay shallow now when does a function exist when does a limit exist sorry a limit exists in three conditions we need to know three conditions what are those three conditions first and very very important.
If left hand limit is equal to right hand limit and see a lot of people they'll just say this wrong when left hand limit they'll say if a question comes limit exist what are the conditions okay they'll directly check only left hand limit lhl and they'll check for rhl if these two are equal they'll say limit x is strong.
If these two are equal and that is equal to a finite value then limit exists see if my lhl what is it lhl is equal to rhl and that is equal to some finite value k then i can say limit exist then i can say limit exist now if that is equal to some infinite value if lhl left-hand limit also goes to.
Infinity right limit also is tending towards infinity so can i say limit exist no because again infinity we have no clue no again infinity we don't know for me let's say infinity is five million dollar for somebody else it really said six million dollar lucky will say eight.
Million dollar goku will say sir ten million dollar so infinity is a what a perspective term this bigger term you can come up sir let's make this bigger term the most somebody will say sir add 10 it will be more bigger than that so like that we can't have an infinite number we don't.
Know what exactly infinity is clear so that's why if my left hand limit and right hand limit are equal and that is equal to a finite value then we say limit exist clear everyone the condition everybody say with me left hand limit equal to right hand limit equal to finite value limit exist if these three are followed.
Then limit exist come on everybody very simple thing okay and function also will check like this left hand limit equal to right element equal to finite value equal to finite value clear clear everyone yes no yes no in the chat section come on fast slowly slowly i want the what is your students to be a bit faster on the.
Chat part okay shallow pavitra okay clear next everybody so for limit we got the condition that left hand limit is equal to the right hand limit and that should be equal to the finite value then only we can say limit exist if it is anyway.
Equal to infinity directly we can say limit doesn't exist so that time don't say lhl equal to rhl okay that time don't say lhl equal to rhl your lhl has to be equal to rhl and that has to be equal to a finite value okay still here everybody clear where limit exists clear now coming to the next part.
Now what does lhl means left hand limit so left hand limit means what let me explain you that so let's say this is a function i want to find the limit at a point okay i want to find the limit at a left hand limit means value which is just smaller than a left hand limit means value which is just smaller than a or we can we are.
Approaching this function from the left hand side or we are approaching the function from left hand side so let's say this is my graph this is 2 i want to find the limit at 2 so left hand limit left hand value will be what somewhere just approaching to 2 but smaller than 2 so 1.999 okay.
Right hand means from the right hand side we are approaching so value which is just bigger than just bigger than 2 so that's why left hand right we write a on top we write minus a lot of people they do mistake that a this minus when i write limit x tends to 2 minus f x.
It doesn't mean student it doesn't mean that what is a limit we are talking about limit x tends to what is a minus 2 no a lot of people they'll put value directly minus 2. wrong this limit x tends to 2 minus means what we are approaching to the value 2 from the left.
Hand side means a value which is like 1.99999 like that clear clear everyone because 2 plus they don't do mistake 2 plus and plus 2 same thing but 2 minus they'll be doing mistake clear so don't do this mistake don't do this mistake clear everyone will go for the next page yes no yes no fast very simple we'll end up very fast don't.
Worry next here same thing i've explained so here i've said lhl limit exchange to two plus mins values which are just bigger than two like 2.001 and limit x tends to 2 minus means values gesture which are just smaller than 2 clear same thing whenever we say talk about left hand.
Limit we are approaching from the left hand side and right hand limit from the right hand side that's it clear now everybody try a very simple question here limit x tends to 2 x square minus 4 by x minus 2 x is not equal to 2 try everybody fast fast fast.
Run clear good keshev not there you there keshev karthiker hi sir okay done hello everybody so everybody got the answer since x is not equal to 2 what we can do limit x tends to 2.
X minus 2 x plus 2 correct x minus 2 now first we'll cut because x is not equal to 2 okay and now we can put easily put the value so it will be 2 plus 2 that is equal to 4. now wrong it won't be 0 or 1 second.
It won't be four on it won't be four okay yeah sorry it will be four it won't be zero how did you put zero because see the moment i put zero zero by zero is not defined okay so we can't do that so first we'll cut the this term x minus 2 and x.
What you say this one x minus 2 and x minus 2 and then we'll get the function as limit x tends to x plus 2. now we'll solve for this function clear in this we substitute the value clear shallow.
Got it run r1 again here run see in this function limit x tends to 2 x square minus 4 by x minus 2 if i put 2 now i'll get the form 0 by 0 which we can't use it okay 0 by 0 is not defined form so we can't write that is equal to 0 okay.
So what we're gonna do we'll first simplify so if i simplify on the top i'll get x minus 2x plus 2 divided by x minus 2 i'll cut these two now my function will be x plus 2 limit x tends to 2. now i'll solve i'll put the value so what i'll get i'll get 4 clear yes you can definitely write x minus because see a square minus b square.
Formula is what a plus b into a minus b okay now let me tell you uh one story we'll come back to this okay one simple story uh this zero by zero what do you think what will be the answer this zero by zero a lot of students what happened is.
One kid was there if you all remember in his i think so fifth class or seventh class i think the fifth class okay in his fifth class when the teacher was teaching him about the divisions okay they were teaching about the divisions and unitary method so he asked a question he said sir.
If uh he was the teacher was giving an example that if you have 10 bananas and you have 10 friends so how many bananas will be getting per friend so a lot of student every student said yes one okay now.
He asks sir what happens if we don't have any bananas we don't have any bananas and we don't have any friends we don't have any bananas neither we have any friends so the number of bananas is zero and the number of friends is also zero in that case sir.
What will be the answer now the whole class started laughing the whole class started laughing they thought what kind of question is he asking okay it's like what'd he say he's one of the dumbest person in the class okay so anybody knows who is that guy.
Anyone knows anyone knows who is that guy who asks this first question nobody come on come on you all have watched the movie correct the man who knew infinity yes or no nobody nobody has watched the movie the man who.
Knew infinity lucky it was a ramanujan sir okay so ramanujan sir was the first person he popped this question popped in his mind that if we don't have any item to give and neither we want to give it to anybody in that case scenario what will be the answer.
Okay it was ramanujan sir the greatest mathematician of all time okay his math level of superiority in maths is way beyond anybody in the whole world okay so even people like mark zuckerberg they have come up and they have said that very few genius people are like what is it ramune sir.
Nobody is that genius like ramanujan sir okay so it's very rare part that somebody somebody is that genius clear if you after the exam if you all get time do watch what you say the man who knew infinity very good movie okay but.
We'll learn a lot from it okay cello let's see now same thing just i wanted to explain if the left hand limit is approaching infinity here if i am finding the limit at zero left hand limit is also going where infinity right hand limit if i come approach from the right hand side to this function also going towards.
Infinity so left hand limit infinity right and limit infinity can it be limit exist no because infinity is not a finite number so limit won't exist limit won't exist don't do wrong okay you like so left hand limit also going infinity right hand limit if we approach from the right hand side also going infinity so left.
Hand limit equal to right hand limit directly limit exists no clear clear got it got it everyone come on come on come on yes no yes no yes no cello.
Now let's see the next one sometimes we use uh this form also to solve the limit part like limit whenever limit x tends to a minus is there f x we can write it as limit h tends to 0 means very very very small.
Number and from a we have subtracted this h so basically what we're trying to say see if i have limit x tends to 2 minus so basically i'm talking about the number which is like what is say a 1.99999 how do we get it.
We get it by limit taking a number which is almost like point zero zero zero zero one okay and that is almost tending to zero we have taken this is my h so f of two minus point zero zero zero zero zero one so i'll almost get here what 1.9999999 like that so same thing just.
The way of writing whenever i write a minus h it means what i'm taking from a i'm subtracting a very very small number so that i can show that i am approaching the value a from the left hand side and the moment i add a very small number it means it shows i'm approaching this function from the right hand side clear everyone.
Clear everyone this also i've again explained limit extends to 2 minus is not equal to limit exchange to minus 2. a lot of people you do mistake try not to do that okay whenever you see limit x tends to 2 power minus here okay and directly a lot of people they use like this it is wrong it is wrong.
Clear everyone clear this part very simple nothing is here to layer got it yes no yes no come on fast fast yes no students clear anybody any doubt anybody has any doubt no.
So if you have doubt now try to be a bit more faster just say it set out if not i'll move on okay because we want to be a bit more faster we have to complete today okay goku yes no doubt okay yes you put it up for no doubt and no means you have doubt.
Clear so that we are able to understand because a lot of times no means also no doubt yes means also no doubt so it becomes difficult or you can write simple nd no doubt shallow next page now very very very important everybody write it down.
Hi srisha everybody write it down higher on tell me these are the seven indeterminant form which we don't know the answer till now which we don't have the answer till now and whenever our limit gets stuck in these forms we use.
The other methods to solve it clear so what are these forms the first one is zero by zero the first one is zero by zero clear just know what rom origin sir said if we don't have anything to give and we don't have any people to give so in that case still confused in the first question.
Okay no worries ron when i'll be taking the question i'll again come back to that don't worry okay since you have already informed it will be there in back of my head okay x minus two k a square minus b square formula 1 x square minus 4 what is a 4 was there.
No so x square minus 2 square so a square minus b square formula camera 1 x minus 2 x plus 2 clear cello next is what infinity by infinity if i'm dividing one big thing by another very big thing.
So what will be the answer we don't know because both the big you can't quantify you can't justify same thing in the case of minus now how will you give priority understand here will it be positive will it be negative yes or no tell me how many people tell me guess this answer.
Infinity minus infinity what will be the answer will it be positive will it be negative will it be zero come on let's see tell me how many people say is positive raise your hand how many negative raise your hand and how many are going with zero infinity minus infinity let's see how good you can think everybody fast fast.
Fast sunday said not defined what about others you want to go with sanders you want to go with something else so you have both the options kids let's see cello character fast zero one said zero what about others.
But answers put answers shirisha gokul pavitra zero zero zero three people said zero what about others i want everybody to answer say understand then only you'll be able to remember in your what is your mind for a longer time it will be easier.
Otherwise you mug up then it will be difficult nobody come on come on karthikey lucky keshev no answers from you guys come on hello yes no yes.
Everybody how can it be zero infinite can infinite can be any number yeah we'll be discussing some of this let's see the other answers what about other answers come on fast fast fast karthikey not defined okay so see here students we don't know whether this infinity is like.
Whether this one is big or this one for somebody let's say this infinity can be one trillion dollar now the some other body will be say sir one trillion dollar and one rupee so we don't know he'll say sir one trillion dollar and two p mine one so we don't know which infinity is bigger.
Which one will dominate okay so that's why these all are not defined these all are not defined parts okay these total seven indeterminant forms are there so these all are not different and not defined clear.
Shallow so one is zero by zero everybody remember zero by zero infinity by infinity infinity minus infinity zero into infinity y you'll say sir zero has a property whenever you multiply with anything with zero it makes it to null it makes it to zero what about.
Infinity infinity has a property you multiply anything with infinity it makes it infinite so which property will dominate well zero dominant will infinity dominant we don't know so again this is not defined clear zero power zero anything power zero is one if i write x bar zero it'll say sir one what about zero power zero again we.
Don't know so it is not defined clear same thing is infinity power 0 clear and 1 power infinity so anything power infinity becomes infinity but we know 1 1 1 1 1 if we keep multiplying one power in one has a.
Power that it always remains one so now again we are in a stuck situation which one to choose so these seven okay these seven are the worst enemies for you these are the indeterminant forms where directly you can't apply the limit now for cet and comet k for cet and comet k.
These two will be mostly coming in zero by zero infinity infinity 0 by 0 and freedom by infinity keep in mind 95 percent of your questions will be from 0 by 0 infinity 0 0 infinity by infinity clear and there you just need to use one rule that is l hospital rule l hospital rule okay some power once will come will show how.
To convert it very rare it comes power to infinity also will come will try to convert it into this part don't worry clear shallow now anybody knows what is l hospital rule l hospital rule we write l hospital but we don't say l.
Hospital s is silent so it is l hospital okay okay we'll study about l orbital very soon now very simple write down these formulas along with me write down this formulas fast come on everybody write down these formulas.
I'll just give you three minutes not more than that alright and fast one minute i'll be back okay try this for students let's see let's see let's see come on so we are getting a varied answer for this what is the answer for this everybody i.
Think so the first one is damn simple if i put 0 sine 0 is what 0 and cos 0 is what 1 so 0 plus 1 1 only very simple now what about the other one here.
A lot of people see first we cannot put the value can we put the value directly yes we can put the value directly okay we can cancel it also and put it so what we can do if i put it what will be the answer now tell me everybody will i cancel and.
Put will i put it directly will i put directly or will i cancel and put what do you say see even if i put here 2 it is not coming in the form of 0 by 0. so i can directly put or if i want i can just use.
This formula a cube minus b cube cancel it and then put it so in both the ways will i get the same answer yes no kids tell me fast so the first way what i'm doing is i'm cancelling making it into simple aq minus b q form a minus b into a square.
Plus a b plus b square correct yes or no because i know x is not equal to 3 i can cancel so what i will be getting here tell me and directly if i'm putting 8 minus 27 by 2 minus 3 what i am.
Getting 19 clear everyone whichever way you go it will be the same thing got it yes no yes no in the chat section fast come on come on so these two properties were just to show how we were applying this formula.
Where this function sign i took it one part because i took it separately so don't get confused whenever plus or minus is there we can just use limits separately if sometimes they'll give you a question limit x tends to 0 2 sine.
X what are you going to do just take this 2 outside 2 limit extends to 0 sine x solve it whatever the value multiply with 2 clear now if two functions are in multiplication if two functions are in multiplication so limit x tends to 0 sine x into cos x what are you gonna do.
So i can do it separately like this limit x tends to zero first i'll do sine x into limit x tends to zero cos x clear everyone in multiplication case also i can separately put the limit into the both the functions and solve it clear don't get confused we are just using the formula a q minus b cube root.
A minus b into a square plus a b plus b square clear just use the formula you'll get it now very very important part okay a lot of times you'll get limit solve this question let's see how many people are able to solve limit x tends to.
Zero log of tan x tell me the answer and number 2 limit x tends to try this too come on come on fast kids fast we'll.
Start the timer also everybody fast i've just given simple questions on on the application part so that you get the grip of the even if you haven't done the chapter forget about you get the grip put down your answers put down your answers for zero second zero now for everybody who is attending the.
Session right now do understand that one second what about others others others no answers from keshav what do you say sandesh lucky hi naveed vimal after long time answers.
Nishat now telling the answer students okay now see so we should understand we should know about a bit about the log also okay so whenever i'm putting so this can be written as limit x tends to 0 correct tan x and i can take the log outside.
So if i put tan 0 i'll get 0 only and i know log 0 is not defined so the answer for the first one will be what not defined same thing here also limit x tends to 1 cos x log.
Now cos 1 is how much student cost 1 is how much cos 0 we know it's 1 what about cos 1 it will be cos 1 only so here it will be what why people wrote this one as 0 or not defined what is the value of cos 1 degree just value not defined for both how.
See cos if you say cos can take any value correct let me show you one second this is the graph of cos x at 0 it is 1 correct now this is my pi by 2 pi by 2 is how much.
Pi is 3.14 divided by 2 it's bigger than 1.5 so somewhere here 1 is there so this is cos 1 value so in that case what will be the answer in that case what will be the answer one degree of one radiant now let's say here we can put.
In terms of radian also because we are talking about radian just one half put the value so even if it is one radian this one is how much pi degree is equal to what if i want to write in the radiant form what will i get the value.
Pi radian equal to 180 degree we can convert and we can go correct the main point here is that one we can easily put up here even if it is one radian log let's say i don't know the value but i know i can substitute one radian yes or no.
Yes or no public come on yes or no we can put here correct so will it be defined undefined tell me a lot of people they wrote in the answer undefined will it be undefined check it fast srisha mam will it be undefined or it.
Will be defined come on come on people are able to understand see whatever the value here okay cos 1 is exist okay cos 1 exists because the domain of cos function is all real numbers.
So i can put one let's say i'm talking about radian only i'm putting putting one so log of cos 1 will exist so it won't be undefined correct so it will be defined i don't know the exact value but it will be defined okay shalom everybody got this one.
Okay everybody got it anybody has any doubt got it everybody yes no yes no fast don't worry while solving the pyq will again go back okay we'll again understand let's see the next very simple just write down fast this formula.
One result this one and the another result this one once you have written the result i'll give you one example to solve and very simple example what is the result telling whenever we have limit x tends to a x power n minus a power n divided by x minus a so it is directly.
The derivative formula n into a power n minus 1 clear so let's see an example limit x tends to 2 x power 10 minus 1 0 to 4 by x minus 2 what we can write we need to write this in terms of powers of 2 so limit x tends to 2 x power 10 minus 2 power 10 by x minus 2.
So if this happens it's similar to this so that is equal to n will come so what is my n 10 10 into 2 power 10 minus 1 so 10 into 2 power 9 so 2 power 9 is what 5 1 2 and then 10 so 5 and 2 0. clear check it very simple formula we already know this just for showing you all i.
Showed it clear anybody any difficulty in this the first formula anybody any difficulty in the first formula kids it's just a formula we need to apply okay now coming to the second formula what is.
My second formula limit x tends to 0 1 plus x whole power n minus 1 by x that is equal to what n so similarly they'll be giving you a question limit x tends to 0 1 plus x whole power 5 minus 1 by x so directly what will be my answer.
The constant term 5 clear everybody clear no doubt write it down fast very simple formula once you write it down once you practice it will be done right down right and right and fast clear clear clear everyone the only formulas which are needed i'm.
Giving you here some one or two extra i've given so that you don't have to worry for ct and comment k okay i have not included any formulas which will be important for je or others okay cello everybody written yes no written yes no i'll go fast no sir no condition wrong.
Okay last question see last question we just uh which question this one seen this directly will have a formula it is basically this question only no problem number two or you want to ask the previous one.
Written no doubt but sir explain last question later okay last question i'll explain later no worries just remember this formula this is directly the application if you remember this formula this is directly the application okay or log voila okay no issues i'll again explain in the last part for those who.
Are having difficulty with the log uh forget about that right now or once the session is done i'll again explain for those who have doubt the log one okay shadow now again trigonometric limits trigonometric limits very very important a lot of people we forget and we don't have formulas in one place so that's why.
I've jot down the formulas nothing more than that just write down fast and finish it out whether it is sine x by x or x by sine x same thing tan x by x or x by tan x same thing don't get confused sine inverse x cos x all formulas i've written just write it down once fast.
And once written two tell me so what i've done because a lot of times students say sir we don't have limits formula and some other book is giving so much a formula some less so whatever the formula is required for ct and comet k.
Okay from the year almost around to 2017 or 2014 to 2021 each and everything we have kept so that nothing goes here and there okay clear right on fast i'll be back in a minute write down fast done everyone done everyone done what about others.
What about others i'm thinking everybody has written very simple very easy formulas just so that the formula everything remain at one place so i've given it okay now here students okay note down the formula but don't worry about these two formula.
Directly you won't be able to understand right now while solving the question i'll explain don't worry just write down these two formula these formulas also you write it down i'll explain while doing the questions okay directly normally we won't be able to understand like this.
But we'll see a lot of students they said sir what if a question comes difficult one part infinity or other questions so how we gonna do and use this formula i'll show you where f x can be a function like 1 power infinity or 0 power infinity or infinity power 0 so a lot of things are there how to solve that kind of questions will.
Show you that type just write down the formula okay that's it one moment you write it down do tell me okay fast fast fast suddenly the screen went off don't worry came back fast write down fast right and fast once done do tell me.
Okay so none of the formulas outside this domain will come okay so don't worry just write down these formulas and each and every question of limits you'll be able to solve it i'll show you how to apply also don't worry so just write down before so that we have all the formulas.
Required to solve it pavitra done very good what about others of shirisha nishat uh okay let's see in the next one last set of formulas okay one more is there.
Fast fast fast now here if you're not able to understand no issues just write it down with the questions we'll be able to understand very simple very easy these are again for the ones which is like one power infinity form.
How do we do it how do we solve it so do write it once i'll again explain with the example only okay because without example it will be a bit difficult students okay to understand so we'll see the example how we are working out how we are doing it you'll be able to understand in a very simple way nothing is there in this.
Done everybody shallow everyone done can i go to the next page then everybody fast fast fast others sanders only written i'll move to the next page students okay.
Now again this result i sinapa see limit x tends to zero whenever we have one plus x by one by x okay it's basically it's basically 1 plus 0 by 1 by 0 so that is basically one power infinity form and this value is e.
Okay now if somewhere lambda comes so it will be e power lambda clear now if i want to reciprocate here and reciprocate here these two just have interchange here it was 1 by x here it.
Was x this time i'm writing limit x tends to infinity 1 plus 1 by x it means again see 1 plus 1 by infinity is what 0 so 1 plus 0 basically 1 and on top we are writing infinity so basically it is of the form 1 power infinity so the value is e only the value is e only.
And whenever we get lambda extra we can put e power lambda clear same thing is there nothing more than that clear write down fast write down write down fast we are almost done we are almost done one last yeah l orbital done.
So we are almost done whatever wherever whichever top book you take it okay and any questions which are coming from cd or comment k these formulas of the limits is more than enough and sufficient to solve each and every question application i'll teach you don't worry for that but formula should.
Be in one place so that's why i have given so that you don't have to go like this book searching so this formula is not there that formula is not there so that's why i gave it in one place one go so that you don't have the trouble asking formula sheet you have formula page have you prepared okay so that your revision part becomes easier.
And everybody is memorizing and revising the formulas yes or no is revision going on children yes or no keep the revision going keep the revision going very very very important okay the more and more chapters will be taught the revision will be difficult i know.
But keep it going keep it going don't leave it then everybody done everyone and do write down in the chat a bit faster students done yes what about others what about others come on come on come on fast.
Can i go for the next page students okay done now that was not needed this formula again two standard results very very important very rarely they have asked but important if in case ct and comet k they're going in kgf mode they can ask this formula okay so in that case.
Do it otherwise very rare it is needed i'll show you the short trick how to do it without this also but sometimes in some questions it is required so do write it down fast just two more formulas so each and every formula or whichever is required even i think so for the j means also you won't get more than this.
Advanced few things are there we haven't done so but mains also it can help for the formula part done done students we have less time we have to finish the chapter and then start with the pyq oh it's.
Alpha any constant alpha not infinity cello students let's see now the very and very and most important formula do mark it as very important l hospital understand this okay because this will be using in 90 to 95 percent of the question and very simple this part this for what is a formula or.
Technique is very very very simple what does l hospital says is if you are getting a limiting question in the form of 0 by 0 or infinity by infinity so directly what you can do you can differentiate the numerator and denominator separately till you come out of this form.
Till you come out of this form and when you differentiate kids a lot of people they do mistake what they do u by v allah we are not doing u by v numerator separate denominator separate numerator separate denominator separate let's understand like let's say we have.
Limits x tends to zero sine x by let's say tan x so if i put here student if i put a 0 what i'll get sine 0 by cos 0 tan 0.
Sine 0 is also 0 tan 0 is also 0 so it is coming in the form though we can do more simple form than this for this question but it is coming in the form of 0 by 0. so here what we can do according to l hospital i can differentiate numerator i can differentiate numerator and denominator separately what is the.
Differentiation of numerator sine x what is d by dx of sine x students come on fast write down in the chat section d by dx of sine x fast come on students d by dx of sine x add on fast let's see how many you remember derivative formula d by d x of sine x students.
Nobody come on derivative of sine cos x and derivative of tan is it cos it's minus cos a lot of people still confused ah.
Cos minus cos is just cos okay sinapa is just cos clear now derivative of tan secant square correct so what will be the answer what will be the answer so what will be the answer.
Limit x tends to 0 1 by secant cube x can i write this or we can write 1 by secant square is cos cube x correct so cos 0 is what 1 so the answer is 1 yes or no clear now i could have done another method.
Also for this question but this is to explain you how we are doing l orbital this is to explain how separately we have differentiated sine we didn't apply u by v we didn't apply u by v we separately differentiated sine we got cos we'd separately differentiate the.
Denominator we got secant square clear and we can keep on doing till we are coming out of the zero by zero form and we can keep continuing till we can we come out of the zero by zero form clear everyone clear everyone yes no yes no in the chat section fast.
Clear everyone clear so 99 of the questions you'll be able to yeah you can do x multiply divide you can do x multiply divide it was sine x by tan x so you could have written sine x by x into.
Tan x by x so this is also one this is also one so you could have done that also in this question that would have been more easy but in other question it will be difficult we'll be coming back to that question don't worry okay clear everyone shallow let's see the next part.
We'll give this example a bit later to you all shalom now some special tricks i am giving and tips directly formulas do write it down whenever we have limit extends to 0 1 minus cos mx they gave you a question limit.
X tends to 0 1 minus cos 5 x by 1 minus cos 7 x what will be the answer directly you can write in the one line form 5 square by 7 square directly clear so these tricks are important do go through and do write it down fast do go through and do write it down fast because this will help you without.
Solving the limits question you can just get the answer without solving the limits question you can just get the answer clear come on come on fast you'll see no youtube a lot of videos are there for this i would say our tricks to solve are also is they're similar this is nothing more it all.
Comes from the concept of some multiple form but right now we won't go the derivation in all that part just keep that in mind how we can use it and simply easily you can directly use it because it will save a lot of time while doing the pyq directly it will save a lot of time you could have converted this whole thing into sine.
This whole thing into sine by using the formula 1 minus cos 2 theta okay or 1 plus cos 2 theta and then you have to do so instead of that directly you can do it uh which one's in upper see these are the tricks in upper if i have.
Let's say limit x tends to 0 1 minus cos let's say 3x by 1 minus cos 4x so directly you can say the answer is nothing but 3 square by 4 square 3 square by 4 square okay that is equal to 9 by 16. so directly you can put the answer clear how it is coming i'm not showing here derivation is there.
It's from this part but i'm not showing because not needed for you all for ct and comedy if you don't have that much time that an exam will sit we'll derive and then do illah we don't have that much of time clear so some formulas will help now this formula if you see limit x tends to 0.
Sine power 5 already same 2x by sine power 5 6x now you'll see the question sir so difficult directly very simple it will be m by n so 2 by 6 whole power 5 so 2 by 6 is nothing but 1 by 3 whole power 5 directly you can get the answer.
Clear so that easy and that fast you won't be able to do if you don't know the trick so these tricks are very very very important it makes you faster for the case of limits guarded sinapa yes no yes no students come on say it in the chat section yes no yes no god.
Didn't get a bit fast a bit fast we can use l orbitals rules also yes yes whenever it is zero by zero form definitely we can use l orbital c theta josie whenever we have 0 by 0 infinity by infinity directly you can go for l orbital.
Anywhere you are getting 0 by 0 infinity by infinity form directly you can go for l orbital also no issues see a limit can be solved in multiple number of ways this sir what do you say one or two ways i'm just showing it clear everyone done everyone will go for the next page yes no yes no fast.
Shallow good i'm going to the next page students okay few formulas of trigo i've just given here if you want you can write it down okay cos 2x is equal to 2 cos square x minus 1 minus 2 sine square x and that is equal to cos square x minus sine square x okay cos mx is equal to 1 minus 2 sine square.
Mx by 2 cos nx is equal to 1 minus 2 sine square and x by 2. these formulas you require sometimes so if you want you can write it anyway we'll be discussing about these formulas in the trigger chapter clear shallow almost done just write down last two.
Formulas more then we'll be starting with the derivative derivative will take 10 minutes we'll finish the derivative we'll take a break at 10 then we'll start with pyq and potential part come on come on fast come on fast fast done everyone.
Let's see the next part now what derivative derivative how do you denote f dash c means derivative of the function at c d of f of c derivative of the function of with respect to c d by dx of the function.
F x that also same thing so these are the ways how do we denote the derivative okay now what does f dash c means f dash means limit basically the slope limit x tends to c f x minus f of c by x minus c if you want to see it in this part.
See f dash c is nothing but the slope slope of a line and that we can do by what tan theta now tan theta is equal to what you'll say sir this is let's say y1 and this is y2 so tan theta is equal to what p by b.
Perpendicular by base and this is x 1 sorry this is taking x 2 this is x 1 so it is what y2 minus y1 by x2 minus x1 now if this is only a function y is equal to fx so basically at this point i'm finding this point is my x this point is my c.
Or you can take this point as c this point as x so basically i'm finding y 2 minus y 1 is nothing but function value at x minus function value at c by x minus c so this is nothing but the slope that is tan theta.
And this is the same thing as the function derivative at that point clear everyone or it went beyond head yes no yes no did it get inside head or it went above do tell me see basically derivative means the slope of the line.
Okay or this tan theta now how can we find tan theta the very easy part of finding tan theta is perpendicular base tan theta is equal to what p by b now what is my perpendicular rate of change of y delta y and what is my base delta x.
Now if my function value here it is f x and the function value here is f of c so can i write f of x minus f of c and here my function value is x uh x values x here and this coordinate of at this point is c divided by x minus c so am i getting the slope only basically.
Yes or no am i getting the slope only is correct are you all getting students yes no yes no in the chat section at least i'll get to know you're able to understand yes no fast fast once again do you want karthikey understand here very simple very easy.
I've taken a lined up this is my function y is equal to f x so this is my point x so if i put in this function x what i'll get here f x and this is my c point if i put it in the function i'll get y is equal to f of c now if i just join this drop a.
Perpendicular from here i want to find this value so what will be f of x because this total is f of x and this total is what f of c so f of x minus f of c now if i talk about this part this is x minus c from here to here x and from here to here is c so this is x minus c now if i want to.
Find the slope theta here tan theta here tan theta is equal to what perpendicular by base what is my perpendicular f of x minus f of c and what is my base x minus c clear everyone check it now check it now but our choice already got it didn't get got it didn't get very difficult concept sir.
Come on come on come on yes no at least even if doubt allah get explained no issues but yes no say everybody got it okay good chello i'll go for the next part physical interpretation physically what you mean.
So let's say a person is running he's running so it's covering a distance okay so the moment we differentiate that we get what speed so the derivative of distance by time is what distance by time means basically the.
Rate of change of distance with respect to time if we find it out what is that velocity okay so if we differentiate the distance with respect to time we get velocity because velocity is what dx by dt rate of change of displacement per unit.
Time clear clear everyone okay now in geometry form what does it mean very very important in geometrical interpretation what does it mean let's say i have a curved up and at this point c.
I'm drawing a tangent so basically f by c means the slope of the tangent to the curve this function at the point the c what does f dash x means f dash x means the slope of tangent at curve.
At point c clear this is the physical meaning if i have a curve and at this point i want to find the derivative of sc so basically what i have to do draw a tangent and find the slope draw a tangent and find the slope so draw tangent find the slope so this theta tan.
Theta is equal to f dash c and that is equal to slope clear yes no clear yes no students clear come on come on come on kids do tell me don't worry questions are damn simple but still tell me yes no clear.
Shallow anybody has any doubt do not hesitate do tell me anybody has any doubt anybody has any doubt we are almost we are done with it anybody has any doubt anybody has any doubt nobody has it out okay cello so we also know the first principle.
Though not needed because this is only needed when you're doing the cbs exam okay then we have plus minus rule we already know product rule you differentiate the first function leave second function as it is differentiate the second leave first function as it is so this is known as product rule.
And quotient rule divide the numerator minus v dash u by v square that's it clear this all everybody knows so this is the end of the chapter limits and derivatives this is the end of the chapter limits and derivatives okay we're almost done with the concept.
Now questions practice so we'll be taking a 30-minute break whoever wants to eat food whatever it is to take it and we'll be back okay so it's a break time any doubt anybody has do tell me okay geometric instead of interpretation see when graph is data one graph is there.
Now i want to find what is the derivative at this point so the derivative of the function at this point is nothing but draw a tangent and whatever you are getting the slope so slope of tangent drawn at point c is known as.
F dash c so f dash c is nothing but the slope of the tangent drawn at point c at points you draw a tangent whatever the slope you are getting that is the f c clear sundesh clear okay anybody else has any doubt lucky.
Vinay hello take a break we'll be back soon okay 30 minutes max don't be here in there hello everyone are you all able to listen properly yes no yes and no so the screen is visible you are able to see the question that's a good point.
Shallow so everybody ready get set ready come on come on fast everyone join fast come on everybody we have to finish limits and then let's start let's start command command two minutes a very simple question on your screen which can give you one million dollar piece come on fast.
Let's see you can solve it you've got late today everybody fast fast fast bye sudeep shallow everybody fast fast fast hi rahul i'll give the answer 0 what about others house how zero roll how you all are getting zero command.
Come on fast come on think come on think it's zero it is not the answer zero is wrong zero is wrong for those who are thinking it's zero it's wrong think on it think on it students sandesh 0 is wrong bro 0 is not.
Differentiation of any number is 0 yes definitely differentiation of any constant number is zero but it is constant number plus a variable so you can't do like that so i'll show the time is up the time is up i'll show here everybody karthike basha.
Pavitra join fast see it loit is there okay gokul is there see one way is you multiply all of them y is equal to multiply all of them so what you'll get if i multiply 1 plus x into 1 plus x square into 1 plus x power 4. so if i multiply just these two what.
I'll get x square plus 1 plus x plus x cube now multiply with these two okay so what you'll get 1 plus x square plus x plus x cube plus x power 6 plus x power 4 plus x power 5 plus x power 8.
Sorry x 7 correct normal multiplication and then what you do and then what you do differentiate it d y by dx so differentiation of this one constant is 0 but plus is there plus is there bro after that 2x plus 1 plus 3 x square.
Plus 6 x power 5 plus 4 x cube plus 5 x power 4 plus 7 x power 6. now put everywhere 1 so what you'll get 2 plus 1 plus 3 plus 6 plus 4 plus 5 plus 7 so 12 10 and 6.
So that is equal to 28 answer that is equal to 28 answer others a lot of people how did you write 0 see if it is 1 plus x dot if i'm differentiating first one is 0 but it is plus here not multiplying so plus and then dx by dx is what 1.
And we can't do directly either you could have done u v w okay directly you can't do these are in multiplication so when i have 3 functions 1 plus x 1 plus x square 1 plus x power 4 i can do u v w that also i can do clear.
Akash sharma sir i'm preparing for cd 2023 what book do you recommend for physics and math for city 2023 first what do you say get the ncrt and exemplar done that's it ncr exemplar done after that again be in contact i'll tell you.
No other books needed by the time we'll be putting up the previous year's papers will be there from our part okay do solve that that is more than enough ncrt exemplar previous year's paper done do not trust the books of aryan mtg and all that questions are not correct okay.
Not always they are putting correct questions a lot of because i have all the books of like mostly five six years along so we have seen a lot of wrong questions are there okay and you don't need anything maths and physics that is more than enough shalom everybody got this everybody got it yes no yes no.
Now you got white shouldn't be zero sandesh rahul done no issues akash work hard any difficulty anytime i'm there you can take the number you can give me a call you can ping me for any kind of suggestion.
Take this number bring it in the whatsapp group whatever suggestion you need you can just put it here whenever i get time i'll give you a call okay cello cell everybody let's see the next one the next question a very simple one let's you can solve it the next question a very simple one you have a two minute.
Timer let's see you can solve it come on fast where is vinay vimal come on guys join fast join fast you have a two minute timer let's see you can solve it very simple questions and for those who are not able to get in the first time.
Once i'm done once you're done with me at home revise it and after revision if you're having difficulty do tell me the next class clear i'll be putting doubt sessions for the what is the crash course also so do tell me in that so these these parts of doubt will work for that.
Come on come on answers public answers answers let's see very simple question let's see you can solve we got the first answer psi memo c option what about others we got the first option c what about others i'll put in the app also whoever.
Hasn't joined till now do join fast we are doing the pyqs of limits very simple very easy do join fast b option a lot of people wrote b option a lot of people wrote b option what about others c and b we got two options what about others come on come on public.
What are the other options karthik no answer we got lohit put up answers right or wrong doesn't matter think of it put answer right so still what time does the session end today's session will go a bit big how much time do you want to say today.
It will be going a bit big more on youtube part around we'll finish this by 11 11 15 and then we have the physics chapter potential and capacitance so we'll be doing that so it'll be a big session tomorrow onwards youtube session will.
Become less and more in the app it will be there okay so now let's see pavitram b let me show you very simple question very easy what we can do here is see what they have given limit first write down the question limit.
X tends to 1 f of x minus 2 by x square minus 1 equal to pi now here if i put here directly 1 i will get not defined 0 in the denominator so what i'm gonna do see i already know whenever i have limit extends to some value.
Some function or top by some function on down what i can do limit i can do the function on top extends to a my limit extends to a the function on down clear so here also i'll do the same thing limit extends to 1 f x.
For constant it won't have any change divided by limit x tends to 1. x square minus 1 correct and that is equal to pi now this whole part i'll just send it that side limit x tends to 1 f of x minus 2 is equal to pi into limit x tends to 1 x square minus 1. now i can.
Put if i put x equal to 1 this whole thing will turn out to be what let do this whole thing will turn out to be what zero so limit x tends to 1 f of x is equal to 0 and this minus 2 was there send it that side so what are you gonna go so what you will get 0 plus 2 so 2 so.
That's the answer everybody got it everybody got it clear yes no clear yes no anybody doubt do tell fast kids anybody doubt anybody doubt good good good nobody has any doubt we can move on to the next question we can.
Move on to the next question cello let's see you can solve this one again let's see you can solve this one two minute timer you have akash god see what about others come on come on fast very simple question very easy ones try loit also got see let's see others karthik see i think so everybody will.
Get c let's see let's see let's see lucky also god see long time no see bro where is yes what is that answer yes is it c or something else.
Nishat not there today let's see anybody has doubt let me show it again if anybody has doubts still so what is my function f x is equal to x bar hundred by hundred plus x power ninety nine by ninety nine plus dot dot dot plus x square by two plus x plus one now.
First differentiate first do f dash x simple so what you'll get you remember the formula x per hundred exponent formula is what d by dx of x power n n into x power n minus one one less so hundred into hundred power ninety nine by 100 clear so if anybody understand this much done.
For this one will be what 99 n into x bar n minus 1 so 98 by 99 plus dot dot plus 2 x by 2 plus 1 plus 0 clear so if you see all these denominator will be cut so what will get x bar 99 x bar 98 plus dot dot dot up to x plus 1 now if i.
Put f dash of 0 the moment i put 0 it will be 0 0 0 all will be 0 to layer only 1 will be left so what will be my answer student c what will be my answer student see sai mama did you get it now sundays did you get it yes no yes no kids now check it check it check it fast.
Check it hi ashes long time lucy all going good got it sanders good iron got late bro clear clear shall everybody got clear hi keshev let's see the next question got a bit.
Late bro okay okay hi yasha's the new one okay now let's see this question very simple very easy try eliminating limit x tends to 0 1 minus cos x by x square let's see.
I've already given you trick how to solve it okay and without trick also you can solve it let's see who can use what akash sharma d not bad not bad akash good keep up the pace d good.
High sheikh smash very simple you can see it is zero by zero form you can use l orbital if somebody is good with trigo they can just use some multiple form get it hi gokul damn good damn good got a bit late bro sandesh d okay.
So everybody is getting d come on come on fast yes no yes no yes no fast any difficulty you can solve it in two methods if you're good in trigo you can use the trigger part one minus cos to theta or in the sub multiple form and if you're not good you can you if you're able to see zero by zero form you.
Can use l orbital differentiate numerator differentiate denominator separately get the answer done the best will be to use l orbital whenever you get zero by zero form or infinity by infinity directly go for l orbital so everybody see here time is up this if i put x equal to zero here what i'll get one minus one by 0 so basically.
0 by 0 form i'll use l orbital so if i differentiate this what i'll get d by dx of course is what minus sine so d by dx of minus cos is just sine and if i differentiate the denominator i'll get 2x so sine x by x is what one so one by two answer done clear anybody anybody has any doubt come on come on.
Fast karthik got a bit late bro work fast work fast anybody any doubt in this l orbital is the game changer bro the moment you see zero by zero infinity by infinity directly nothing here and there directly approach l orbital.
Clear everyone clear clear clear shallow try the next question let's you can solve it try the next question two minutes on the clock two minutes on the clock wrong formula you have to muck up limit sine x by x cube the limit x tends to.
Zero sine x by x value that values done shallow first answer that is c let's see character wrong will get to know come on others fast fast fast run is getting b we got two people answering the question not bad 52 seconds left 50.
Akash answer is five year written pavitra is getting option d okay we'll check will check will check what about others come on come on come on others buckle up fast fast fast let me show you here we don't have that much of time but okay.
First what we'll do we'll write y is equal to this function is given very important question a lot of time this kind of question comes and we are in a stuck situation now what i have to do i want to find d y by dx so derivative of this function so differentiate d y by.
D x that is equal to y dash and that is equal to differentiation of this function f dash of x plus 2 and then chain rule i differentiated this function i wrote f dash x square plus 2 then what i'll do i'll differentiate the inside of that so if i differentiate the inside i'll get 2x.
Clear a lot of people they do mistake here yes definitely track x okay do call in the number given here now so y dash is equal to f dash x square plus 2 into 2 x still here i think so everybody will understand.
What i did i first derivated the function because i don't know anything can be done because i didn't know about the function so just i wrote function derivative and then i differentiated inside the chain rule i differentiated the inside of that so i got 2x now what i need to find i need to find d.
Y by dx at x equal to 1 so put it so that will be f dash here will be what 1 plus 2 into 2 into 1 so that is equal to f dash 3 into 2 so what is my f 3 kids 5 into 2 so answer is what 10 a very important question to all of you very very very important question.
See if you are able to understand see if you are able to understand the option is b option b because see this kind of question is very and pretty much common in cd and comet game so get your basics done very important be very strong in this kind of question.
Nothing is there check it out check it out pavitra clear uh what time you should call today call tomorrow 11 11 am you call in the morning tomorrow or you can call night if you're up you can call around 11 30.
Okay now everybody i'm just giving a similar question okay try to do it instead of this i'm changing the function now this time the question is given as student if y is equal to this function and.
F dash of 5 is given then find the value this one come on do it everybody let's see who got it who didn't get i'll again explain sandesh try this one with the previous ones similar and again i'll explain so that's why i gave a new question so that you can try.
And i'll again explain in that check everybody try try try i'll be giving you two minutes first answer on the screen 35 let's see kids akash again 35 backing the previous 35 let's see what about others two people getting an answer 35 also don't know oh lord third answer 35.
My ma 35 gokul 35 okay if you're seeing a lot of 35 spamming in the chat it's not spam okay okay good so let's see students those who haven't understood okay i'm again explaining understand very carefully so this function i don't know will i be.
Able to open so i won't open it only so what i'll do i'll just write d y by dx is equal to f dash means i have differentiated the function x cube plus x square plus 3. now i've already derivative the function now what i have to do i have to differentiate the inside inside part so.
What i'll get 3x square plus 2x 3x squared plus 2x that's it done same thing so first line very important i just differentiated this y since i didn't know anything about the function so just keep f dash means d by dx of that function into the after that once i've.
Differentiated the function i have to differentiate the inside because chain rule so i differentiated the inside x cubed differentiation 3x square x squared differentiation 2x done now what i have to do i have to find d y by dx at what value at x equal to 1 so i'll just substitute so f dash if i put here 1 i'll get 1 plus 1 plus 3.
And here if i put i'll get 3 into 1 plus 2. so here i'm getting f dash of 5 n2 3 plus 2 is how much 5. f dash of 5 already i gave the value that is 7 into 5 so that is equal to 35 clear everyone yes no yes no everybody clear now yes no come on come on who is not clear do tell.
Me sandesh clear clear bro clear yes or no sandesh clear is it if you still didn't get i'll again explain don't worry after the class don't worry okay.
Once the session is done do stay okay i'll be explaining the questions which is not clear for the other students okay cello got it now very good very good let's see the next question a very similar question to the previous one see ct also repeats comment k also repeats similar type just now we have done tell me the answer.
Fast tell me the answer fast this time and i want everybody to get a correct answer come on come on fast challan singh be in the group b in the group everybody good good good hi lakshmi be in the group okay go cool also be come on come on fast hi my mom be good.
Vimal got the answer be okay very good very simple very easy question okay come on others others be fast be fast we have less time and everybody after the class ends do revise do revise do revise otherwise it will be very difficult to be fast in the exam time your target is 30 seconds not more than that.
Within that 30 second span you have to solve the question okay shallow i'll go for the next question students nobody has any doubt in this hello everybody no doubt no anybody if anybody has doubt do put it up if anybody has doubt do put it up the.
Answer is one wrong answer is one okay nothing is there same thing just differentiate g dash x if you differentiate you'll get 200 x power 199 divided by 200 plus same thing 199 199 x power 198 dot or dot last will be one plus zero so if i put g dash of zero everything will.
Become zero and plus one will be left so answer will be one clear girds on this you are becoming pro so good shallow let's see the next one try let's see who can solve this let's see you can solve this very simple very easy question.
Mostly evening after 7 30. shall we have a two minute timer two minute timer come on fast fast fast let's see how many people saw okay lucky very good okay lucky got this one one or previous one one okay.
Okay okay option one was not there put option one okay correct option one was not there shallow let's see let's see people are getting answer b people are getting b somebody luid got c long time no c bro got late c.
What about others karthik c so this question option c a lot of people wrote option c pavitram wrote b lucky c let's see this question now this question if i see i can solve if i put x equal to 0 i'll get 0 by 0 form i can use.
L orbital correct or if i want i can do something more simple also see here i can do a hospital first and i can do something other also what i can do is here limit x tends to 0 x e power x minus sin x by x now divide numerator by.
And denominator by x or just divide it so what you'll get limit x tends to 0 e power x minus sine x by x now you already know sine x by x is what one and limit e power x x tends to zero is what 1.
So 1 minus 1 student is what 0 so answer will be c clear again see what you have done see first step just divide normally just divide this by this see whenever we get a minus b by 2 can we write a by 2 minus b by 2. so here simple what we did x e power x by x i wrote.
Minus sine x by x i wrote clear limit is there so x x got cancelled so limit x tends to 0 e power x minus limit x tends to 0 sine x by x now i already know the moment i put e x equal to 0 i'll get e power 0 as 1 and i know now x tends to 0 sine 0 by 0.
Is what 1 so 1 minus 1 is 2 and this is what 0 clear anybody any doubt rohan now you got it is it b or c got it students brc everybody got it pavitra clear what about others rohan who wrote be option now you got ytc.
Clear very simple question okay see i could have done l orbital but in l hospital case what will happen i have to do uv so that's why i restricted best is like simple here you can get it clear shallow everybody very good very good very good cello.
Next question let's see let's try the next question previous question i'll again explain don't worry rohan okay this question i'll again explain uh let the class be done okay do remind me okay stay in the call i'll explain now everybody try this question i've already given you a hint i have already.
Given you a trick how to solve it you can solve this question in less than five seconds in less than five seconds i have already given the trick while doing the concept form for those who joined later uh once the class is done go back to the previous write down all the formulas so that it becomes easy so you have all the.
Tricks to solve it in less than five seconds okay chill everybody try fast kids fast so i can see everybody is getting the same value b option b correct very good very good very simple formula.
Very easy formula what is that formula limit theta tends to 0 1 minus cos m theta by 1 minus cos n theta that is equal to m by n whole square that is equal to m by n whole square or m square by n square so directly here 4 square by 6 square so that is equal to what.
We can cut 4 4 6 6 so 2 3 2 three so that is equal to four by nine that is one method clear very good very good if somebody wants sir here also l opportunity zero by zero you can do that.
You can differentiate you will get in that case sine 4 theta by 6 sine 6 theta now what do you do divide whole thing by this by 4 theta this by 6 theta again you will get by 4 square by 6 square you can do any of the methods if.
Somebody want 1 minus cos or 2 theta formula 1 minus cos theta is equal to 2 sine square theta by 2 you can use this also so any whichever you want you can use it and solve it okay clear everyone yes no yes no everybody got it okay shallow next question very simple very easy.
Tell me what will be the answer a very simple question let's see you can do it ron got d what about others akash d see it's basically like signum function it's basically like signal function correct so when x is greater than 0 what you can say limit will open as limit x tends to 0.
X by x so that will be 1 when x is less than 0 it will be open as limit x tends to 0 minus x by x so minus 1 but add 0 it won't be defined add 0 it won't be defined it won't be defined clear so that's where the whole function will go not defined clearance here and crt alloy correct.
Cello clear everyone cello let's see the next one everybody got it yes no yes no chello hi shivani everybody try this question let's see you can solve it come on fast very simple very easy question don't get scared the question is simple try you.
Will get it you need to have a bit idea about the greatest integer function see just keep it in mind the greatest integer functions always gives me a constant value i am the greatest integer function always gives me a constant value okay so based on that you can differentiate.
Just if you remember this point in your brain this gives me whatever it is a constant integral value see whatever the x value constant integral value will get from here so if i differentiate this if i differentiate this i'll always get 0 whatever x is equal to pi by 2 whatever.
X is equal to any other value doesn't matter it doesn't care i'll always get 0. clear just use this you'll be able to solve it just use this you will be able to solve it try everyone come on come on fast people started getting answers c answers.
C didn't get this one okay see greatest integer function wrong is what it is like a function which gives me integer value so if i put 2.3 it will give me integer 2 if i put 3.2 it will give me integer 3. i already explained how it happens if.
I'm doing box of 2.3 so first it will check it lies between which two integers so 2 and 3 now what is the smallest integer among these two so 2 so the value of box 2.3 will be nothing but 2 but see this will give me always a integral value so if i differentiate a constant what i'll get.
Differentiation of a constant is what 0 differentiation of a constant is what 0 correct now i have my function f x is equal to x minus box x minus cos x now differentiate this f dash equal to first differentiation of this is 1 differentiation of this is 0 and differentiation of cos is what plus sign.
Correct so down so f dash x what we got 1 plus sine x now we have to find for f dash of pi by 2. so put 1 plus sine 90 so 1 plus 1 so that is equal to we are getting 2.
Clear the session will end almost now what you say by 10 minutes more it will end we almost done with the pyqs of limits clear everyone so the option is b option is b clear answer is b.
Check it i'll again explain if you have any doubt okay take your time don't be in a hurry understand and then tell me the answer no issues whatever mistake you do it's good but understand from your mistakes okay learn from your mistakes where which part.
Okay so that you don't commit that again cello fast students now how many people have doubt in this question tell me how many people have doubt in this question i'll again explain fast fast fast who has a doubt don't hesitate to ask.
Doubts just do say sir i have a doubt no issues see first thing is you need to understand whether you're doing a mistake or not so there is no shame or nothing like that in accepting your mistakes sir i didn't get i'll explain no issues then you will understand more better.
So do say who god who didn't get okay don't have a shy feeling here we are all here to learn nobody has any doubt lucky no doubt what about others lakshmi ma'am no doubt what about others whoever has a doubt you can put it in the chat section i'll again explain if.
You got it well and good damn good no issues pavitra no doubt understood shivani good okay see the only point here was to understand that this is what our greatest integer function and it always gives me a constant value and if i differentiate it i'll get zero then rest is normal okay.
Clear shallower good now let's see this question this question everybody take your time it is a bit difficult because you need to know about the series also to solve it try if you can otherwise i'll explain try it's not that difficult if you didn't.
Know the serious formula you'll be able to solve only thing is you know you need to know the required formula you need to know the required formula so i'm giving you two minutes time try it let's see yeah class one class will be there in the youtube part regularly and the rest classes will be going on in.
The app for the crash course part everybody don't get scared by the question question is very simple if you remember the formulas it will be done in like this okay just formula part you need to be very proper nothing more than that the question.
Looks very difficult but it is damn simple first you all try then i'll tell you it's very simple you should know two three things what are those two three things you shouldn't remember it first i'm giving you all.
The formula write down whenever i have the formula of 1 cube plus 2 cube the series of cubes first and natural numbers cubes of first and natural numbers the formula is n into n plus 1 by 2 whole square or the same thing you can write it as n square into n plus 1 whole square by 4.
Okay this formula you have to apply on the top okay so let's see how do we solve it limit first forget about this first solve try to go for this so what we'll have limit and by the way this is n one second this.
Is n n tends to infinity okay limit n tends to infinity 1 cube plus i just did that same first part 3 cube plus up to dot dot dot n cube divided by x square ah one second students.
The down also okay we have x no here we have x up to n part one second let me check the question part okay yes this is an only correct the question is correct here this is an only this part we need to first see.
Now they have given everybody okay now i already see here in the in this part what they have given me sigma sigma means summation now what they're telling that if i do a summation from r equal to 1 and it keep going on till n.
And if i put the value here i'll be able to get the value x so what is x first we need to find what is x okay so let's see here everybody now what you're doing these kinds of cases don't get scared first put r equal to 1 put r equal to 1 what you will get.
2 minus 1 you will get 1 correct put r equal to next time 2 what you'll get and summation is there so that's why plus sign came when i'm putting r equal to 2 what i'm getting 2 into 2 4 minus 1 so 3 next again put r equal to 3.
So what we'll get 3 into 6 minus 1 5 so are you able to see we are generating a and last you put r equal to n so if i put r equal to n so what i'm getting 2 n minus 1 so i'm generating a series of odd numbers yeah these are all kcd questions.
Quesadilla and common k these are all ct and common k questions okay so i'm getting a series of all odd numbers now what is the series of odd numbers if i see now here they want you to know this is a good question if i add 1 plus 3 plus 5 what will i get.
The sum of first 3 odd number that is equal to what 3 square yes or no if you remember this we have done in bt in class 11th okay if i add the first three odd numbers i'll get 3 squared if i add first 4 odd numbers 1 plus 3 plus 5 plus 7.
What i'll get 4 square so if i add so if i add sum of first n odd numbers so.
1 plus 3 plus 5 plus dot dot total n term n odd numbers what i'll get n square yes or no yes or no kids tell me this part did you get it or not everybody tell me this part did you get it or not otherwise i'll again explain just check this part.
It's a good question very rarely it comes up so just do check this part everybody got it or they have a doubt don't do hurry in this it's a very rare one two questions like this so go through once take your time tell me if you have doubt in this i'll go slowly in this okay.
Rest is damn simple once again pavitra ma'am what about others do say it if you want once again last one repeat okay let's take a new page understand this part again very carefully so first i'll go with this part first i'll go with this part.
So what have they have given sigma sigma means everybody say with me summation sigma means summation to power r minus 1 is equal to they have given as x so what they're trying to say us is if i put r equal to 1 what i'll get the first number.
2 into 1 minus 1 so i'll get 1 since summation is there so i have to add and i have to keep adding till n number of terms clear don't worry lakshmi i'll clear it out again so again for r equal to 1 we started and we have to go the loop till n so next.
We'll go to r equal to 2 for r equal to 2 if i put here the value 22 4 minus 1 so i'll get 3. next if i put r equal to 3 students what i'll get 2 into 3 6 minus 1 so i'll get 5 for r equal to 4 if i'll put student 8 minus 1 so 7.
Dot dot if i put r equal to n the last term the summation is going till this is the last term so if i put r equal to n i'll get what plus 2 n minus 1 student and this whole submission if i add all of this i'll get the value x clear still here anybody has doubt.
Till here anybody who has doubt come on come on everybody anyone has doubt in this one till here anyone doubt till here yes no yes no cello now after this one thing you need to remember in class 11th what we have done if i add.
What you say if i take just one natural number it is the square of 1 only if i take first two odd numbers first two natural odd numbers it is a square of two if i take first three odd numbers it is the square of three if i take first four odd numbers.
Summation of first four odd numbers it will be the square of four now if i take the summation of first n odd numbers kid first n odd numbers seven sorry seven plus dot dot dot up to n odd numbers means 2 n minus 1 we can take so what you'll get n square yes or no what you will get n.
Square so if i add how many odd numbers i'm adding that number square i get how many odd numbers i add that number of squares if i add two odd numbers i get 2 square if i add 3 odd numbers i'll get 3 square if i add first four odd numbers i'll get 4 square clear everyone till here yes no be calm.
Have patience you will be able to understand okay clear till here tell me are you all clear till here yes no yes no everybody is clear to here if anybody has doubt do tell me i just didn't do anything i just put the.
Value of r i made the series then i saw that if i add the first two natural numbers i get two square if i add first three natural numbers i get three square so if i'm adding first n natural numbers i'll be getting n square that's it okay shalom so what i got x is equal to n square.
So can i say if i want to find x square that will be equal to n power 4 correct so x square will be equal to n power 4 now come to the limit part now come to the limit part so limit n tends to infinity this is n here n tends to infinity 1 cube plus 2 cube plus.
3 cube plus dot dot dot up to n cube divided by x square so what we'll get limit n tends to infinity now use the formula of the cubes what is the formula of the cubes i gave what is the formula n square into n plus 1 whole square by 4.
So n square into n plus 1 whole square by 4. now x square is nothing but n power 4 so this n square gone gone clear so what we got limit n tends to infinity n plus 1 by n whole square into 1 by 4 correct everybody.
Now what i'll do i'll try to make it more simple limit n tends to infinity i'll divide am i getting 1 by 1 by n whole square into 1 by 4 now if i put infinity 1 by infinity is what student 0 so what i got 1 into 1 by 4 so it will be 1 by 4 clear everyone 1 into 1 by 4 so that will be.
1 by 4 will be the answer clear rowan you wrote a wrong ass answer bro what about others check it check it fast got it everybody do practice this again i'll explain don't worry again i'll explain this is a good question but do.
Practice because once you practice then only with revision it will stay because see a lot of students are there therefore doing the crash course and they haven't studied like there is a long time gap at the 11th so a lot of formulas you'll forget a lot of things you'll forget but since you.
Have made up your mind that you'll study you'll work hard in this one month so all you have to do even if you're not getting fully give time afterwards once you take a break against it again understand okay and then slowly you'll start getting it.
So don't give up that easily okay if you want to rank less than thousand don't give up that easily from n square part see after n square just come to the limit part if x is equal to n square then x square equal to n power 4 okay now if i do the lcm of this and then put the value of 1 cube.
Plus 2 cube plus 3 cube plus dot dot n cube i'll get n square into n plus 1 whole square by 4 and down x square means and power 4 so this n square and then power 4 will be cut we'll get n square down so what we'll get now what i did is 1 by 4 into n plus 1 whole square by n square so.
Since both are square so i can just insert inside so that is equal to n plus 1 by n whole square i can take so that only i solved 1 by 4 into n by n i wrote it as 1 plus 1 by n whole square now 1 by n means what 1 by infinity means for 0 and this one as it is so this whole thing will be just one so one.
By four into one so that is one by four clear yes and this i'll be giving you one more question like this type don't worry okay i'll be giving you all one more question like this type don't worry okay and your test series will be released on 25th for those who have done the.
Preparations join the test series and uh get your ranks see where you're good where you're bad what you is going good okay hello let's see the next part and for those who didn't get still now.
Again at the end of the session i'll again explain don't worry so let's see the next question this question you can solve in less than not this one first yeah try this one first this question you all can solve in less than 30 seconds try this one you all have the energy to do the.
Physics class after the maths class yes or no put it in the chat section i will take a break or 12 will sit for physics we'll take a break 12 will sit for physics now everybody got a good idea of the limits if you solve the p by q it will be done.
If you again resolve the previous year's question you'll be done clear shallow so lakshmi ma'am you got b what about others uh rohan got d lohit got d let me show it to you all.
Okay so almost done times up so let's see so understand very simple very easy the question is what they have given limit x tends to 0 tan x by this part so if i put 0 tan 0 0 and down if i put 0 4 square root of 4 minus 2 so.
That is again 0 so it is 0 by 0 from everybody yes or no the moment you see zero vector form what do you do l orbital the moment you see 0.04 what do you do l orbital dam simple dam easy what are you going to do try a so it will do limit.
X tends to zero in l orbital what we do differentiate the numerator first so tan x differentiation is what secant square x and if i differentiate square root i'll get one by two root under of 2x plus 4 then differentiation of 2x just 2 done so this 2 this 2 got cancelled now if i put x equal to 0 6 0 is what 1 and.
If i put here 0 i'll get 2. so answer is what sorry this is 1 by 2 so answer is what two that's it was it that difficult students was it that difficult students how are you getting four karthikey how are you getting four how d is correct bro.
How karthik how are you getting four somewhere if we missed anything secant square x we got 1 by 2 root number of x plus 4 so if you put it up if i put x equal to 0 nope bro take a chill pill don't go so fast.
Now the answer is damn simple the moment you spot the moment you spot zero by zero go with l orbital differentiate numerator separately differentiate denominator separately done clear everyone clear everyone shallow.
Everybody try this one last question let's see i'll increase it this is the last question for the maths 2021 i'm giving two minutes here we are getting a lot of answers in the group b b b everybody getting b.
What about others come on come on fast everyone is b anybody has any doubt in this just put the value check the answer put the value check the answer correct very simple question 2021 paper damn simple damn chill nothing was there in the question correct.
So everybody got this answer got it everybody shallow do you want me to explain anybody be good so limits we are ending the chapter done clear anyway everybody clear everyone.
Will be taking a 30 minute break relax your mind relax your brain after that we'll be going for physics potential and capacitance potential and capacitance a big part of the concept is left we'll finish the concept part and then directly we'll go for previous year's questions okay.
Clear cello work hard kids if last only one month is there to get a decent score damn good score for your college work hard because this last one month you can change your score by anything and get into the whichever college you want and then chill next within next half an hour we'll.
Start 12 max by 12 12 5 i'll be there okay so 12 12 5 i'll be there 12 am today 12 am we'll be having the physics class potential and capacitance we have done a part of it we'll finish the chapter potential and capacitance so till then.
I'll be right back hello hello hello hello join fast fast fast okay oh join fast everyone come on come on fast sure karthikey will tell the tomorrow's timing.
Tomorrow today only it's already 21st b hello everyone if you're done if you're too tired we can have a class morning seven to nine or like six two let's see what do you all students say.
A class now from like 12 15 to 2 or better to get up in the morning 6 30 to 8 30 or 6 30 to 8. what do you say which one which one you want to go for because kids are almost done.
Okay so what do you say tell me fast fast do you want me to end the session now it's almost and instead of night session we'll have a morning session tomorrow morning you want to have a night session a morning session tomorrow lucky morning.
So ask please reply sir what a reply about what's worse karthikey okay with both so since others are not there low it morning session so you all will be up no okay don't leave me alone in the morning okay 6 30. i'm seeing i'm up nobody is.
There so everybody put down the alarm for six okay and get up okay so i'll see you all in the morning session 6 30 to 8 30. see you all in the morning session 6 30 to 8 30 okay don't be late okay so we'll keep be keeping the class still.
Here doing for mothers also i'll put up in the app morning session 6 30 to 8. clear cello then we'll keep end the class here morning 6 30 a.m session okay.
Cello okay then we'll end the stream then you