18 Calculus Part 4 Limits

追加: 2026-04-29

[00:00:00]okay so our next concept tips and this you can say is our first concept from this sceptred calculus till now we were discussing some of the basic concepts that we will be needed throughout this chapter we have learned how to draw a graph of different type of function and we have seen how to do graphical transformation so now our next topic is this limits okay this limit is very very important if you want to understand the other the next two topics continuty a difference ability so the idea of limit I will explain with the help of one basic example suppose if FX is excess power minus 4 by X minus 2 then as you can see that this function is clearly not defined when this denominator becomes 0 that is when X is equal to 2 okay when X is equal to 2 then you will get 2 minus 2 0 in denominator and therefore at 2 this function is not defined okay but if we want to check the behavior of did this function when X is close to 2 okay then then we use this concept of limit at 2 the function is not defined but when X is slightly greater than 2 or slightly less than 2 then this function is defined okay so we want to find the value of this function f of X when X is very close to 2 then we need the concept of this limit ok now one important thing suppose if we want to approach to this point - okay first of all if we want to check the behavior of this function then we write this thing as limit extending to 2 f of X and this will mean that X is approaching to 2 okay always remember that X is not equal to 2 because when X is 2 then the function is not defined okay if we write limit extending to a then it clearly means that X is not equal to a okay X is either slightly less than a or slightly greater than a so if I am writing extending to 2 then it means X is very close to 2 but there are two possibility X can approach to two either from this left aside in that case the value of x will be very close to two but always less than two numbers like one point a nine one point nine nine and so on so this thing is written as extending to two minus means we are approaching to two from left side okay and second possibly it is that we are approaching to two from right side so in that case the value of x will be very close to two but always greater than two numbers like two point one to point zero zero one and so on so this thing is written as X attending to two plus okay extending to two minus means X is approaching to two from left side which means X is very close to two but always less than two and leave and extending to two plus this will mean that X is approaching to do but from right side meaning that the value of x is very close to two but always greater than two okay so now we are talking to talk about these two things right hand limit and left hand limit what is this right hand limit and what is left hand limit so suppose if X is approaching to 2 from the right side okay so this we will write as limit X as X tending to 2 plus so these are the values of X X is 2 point 1 2 point 0 1 and so on so now we are going to see what is the corresponding value of y from here what are the corresponding value of y for a different value of x so as you can see that this function FX is X square minus 4 by X minus 2 this can be written as X minus 2 X plus 2 divided by X minus 2 now remember that when x is not equal to 2 then you can cancel these two because then in that case these two will be nonzero quantity and it can be cancelled but you have to write here that X is not equal to 2 so this means your function basically is y equals 2x plus 2 where X is not equal to 2 okay remember that this condition is very very important if I don't write this then these two functions are not the same okay this function x square minus 4 by X minus 2 and y equals 2 plus two these two are different functions the reason is that this function is defined for all the values of X whereas this one is not defined when X is equal to two so you have to write here that X is not equal to two now these two functions are exactly same so now for a different value of x what are the corresponding value of y so if X is 2 point 1 the value of y will be 2 point 1 plus 2 so this will give us 4 point 1 similarly when X is 2 point 2 0 1 then value of y will be 2 point 0 1 plus 2 so this is going to give us 4 point 0 1 and you can see that these will be the corresponding value of y for this given value of x right so from this table you can easily see that when X is approaching close to 2 ok this number is very very close to 2 then you can see that the value of y is approaching to a number close to 4 ok when X is approaching to 2 y is approaching to 4 so in mathematics this thing is written as a limit X tending to 2 plus 2 plus means we are approaching to 2 from right side so limit extending to 2 plus f of X is 4 because when X is approaching to 2 from right side the value of y is approaching to 4 now this is nothing but your right hand limit right hand limit means X is approaching to this fixed number 2 from right side ok so and the and the meaning of this statement is ok we are writing this thing then the meaning of this thing is this thing is rated as when X is approaching it or 2 from right side the value of y is approaching to 4 remember that neither X is equal to 2 nor Y is equal to 4 ok X is very close to 2 and Y is very close to 4 so this was all about right hand limit now we are in the position to talk about left hand limit left hand limit means X is crossing to two from left side okay meaning that the value of x is very close to two but X is always less than two so now what will happen we have again again the function is same y equals two x plus two x not equal to two so when x is 1.9 you can see that the value of y will be one point nine plus two this will give us three point nine when x is 1 point nine nine then the value of y will be one point 9 9 plus 2 so this will be three point nine nine similarly you can see that these are the corresponding value of y for the given value of x ok so now again from this table you can make one very interesting observation that when X is approaching to 2 from the left side okay when X is very close to two then you can see that the value of y is very close to four okay three point nine nine is very close to 4 so this you can write as limit extending to two minus FX is four okay and again remember that neither X is equal to two the whole point of learning this limit is that at this point the function was not defined and we wanted to see the behavior of this function FX when X is close to this number two so if I am writing limit extending to two minus f of X equals to four so this means neither X is equal to two nor Y is equal to four when X is approaching to two Y is approaching to 4 when X is very close to two Y is very close to four okay and this thing is called a left hand limit if X is approaching to do from left side then it will be left hand limit if it is approaching to two from right side then it will be right hand limit now next comes existence of limit this is a very very important concept suppose if limit X attending to a minus f of X okay this is your left hand limit so limit extending to a minus f of X is equal to limit extending to a plus F of things meaning that left hand limit is equal to right hand limit if these two limits are same again the value of limit that you are getting when X is approaching to a from left side is same as the value of FX when X is approaching to a from right side so if these two limits are same then we say that limit X attending to a f of X exists okay and say the value of these two are same and this is equal to some number L so then we will say that this limit exists and the value of limit the value of limit is this number L so this is a very very important point if the value of left hand and right hand limit are same then we say that limit exists and if these two are different if suppose limit extending to or maybe I can call this as left hand limit now so if left hand limit and right-hand limit are not equal okay you are getting different number when X is approaching to a from different side from left side and right side then this means limit does not exist okay the value of limit will exist only when these two are same left hand and right hand limit are same also once these two limits are same then you can combine these two results left hand limit was two so our left hand limit was for definite limit is for the right hand limit is also fourth so since these two limits are same so you can combine these two and you can write that limit extending to two f of X this will be four okay now here we are not writing whether X is approaching to two from left or right because the value of this limit is the same whether X is approaching to two from left side or right side so it is not necessary to write to minus or two plus limit extending to two f of X is for whether X is approaching to do from left side and all right side but if we were getting different numbers okay for example say in some other problem if say limit extending to 2 minus f of X it's a 2 and limit extending to 2 plus f of X it's a 3 so now you cannot write this thing that limit extending to 2 f of X this will be some number because now the value of this limit will depend on whether X is approaching to 2 from left side or right side okay if because we are getting different number from different direction so in this case we will say that this limit does not exist okay here you cannot write that limit extending to 2 f of X is some fixed number because that number will depend on whether we are approaching to 2 from left side or the right side okay so this was existence of limit now we will take some problems and we'll see how to find whether limit exists or not okay so now we have these three problems based on existence of limit so we have learned that if left hand limit is equal to right hand limit this implies limit exists okay and if left hand limit is not equal to right hand limit I mean this thing is obvious if left hand limit is not equal to right hand limit then a limit does not exist okay so for each one of these three cases we have to find the left hand limit and right hand limit and if those two are same then limit will exist if those two are different then limit will not exist so the first problem is limit X tending to 0 mod X by X so here if we want to calculate this left hand limit then left hand limit means approaching to 0 but from left hand side okay so this I will write as limit X tending to 0 minus mod X by X now remember that mod X is defined in this manner more X is X if X is positive or 0 okay if you have any positive number then you can directly remove them or design for example suppose if I have this number two and since 2 is a positive number we can directly remove the mod sign but if X is negative number and if you want to remove this mod sign then you have to place one minus sign outside okay this will be the case when X is negative for example if I have this number minus 2 and I want to remove this mod sign then I have to write a minus sign outside because if X is negative then minus X will be positive right so here also now zero minus is a negative number see this is your coordinate data this is your number 9 you have 0 here if you are approaching 2x from left side extending to 0 say this 0 minus this is nothing but a number very close to 0 but negative number okay so this number is 0 minus I mean this is not a correct way of writing this because this is concept approaching 2 not equal to any fixed number so but just to explain the idea that 0 minus is a number very close to 0 but with minus sign so zero point zero zero zero many more zeros and then what okay so this is 0 minus so here if I want to remove this mod sign and this X is a negative number then we have to write 1 minus sign here so this means we can write this as the limit extending to 0 minus and if I want to remove this mod sign then I have to write 1 minus sign here so minus X by X okay now these two will cancel and this will give us minus 1 so this means if you have a number very close to 0 then mod X by X will be minus 1 okay and this was obvious because forget about all these things suppose if I simply substitute this number then what I will get and just to explain the idea suppose if in place of X I write minus 0.1 okay so the value of this mod X by X this will be say mod of minus 100 zero zero one and this divided by minus zero point two zero zero one so you can see that this numerator will become positive denominator is negative and both of these are same number so this will give us minus one okay so this means the left-hand limit for mod X by X at x equals to 0 is minus one next we need to calculate the right hand limit the right hand limit will mean limit X tending to 0 plus mod X by X now zero plus means a number very close to zero but a positive number this is your zero plus so if I want to remove this mod sign here then since X is positive number you can directly remove the mod sign so you can write this as X by X these two will cancel you will get plus one again you can think that if I had plus the sign here then mod X by X so this will give us plus one because now these two numbers are exactly same mod X will give us positive outcome X is also positive so the ratio will be one but here mod X will give us positive number X is negative number so ratio becomes minus one okay so here since left-hand limit which is minus 1 is not equal to right hand limit because minus 1 is not equal to plus one so since left hand limit is not equal to right hand limit this means limited does not exist ok so this first one problem based on the existence of limit now we have these two problems these are again very interesting problem because with the help of these two examples we'll learn one important concept so our second problem is we have to evaluate this limit X tending to 0 1 by X so first we are going to calculate this left hand limit left hand limit will be limit X tending to 0 minus 1 by X so this will be 1 by 0 minus okay now here again 1 by 0 minus means reciprocal of a number very close to 0 but this number is negative number okay think of this 0 minus as this number minus 0.001 and zero plus AZ plus zero point zero zero one okay these two numbers so again it is not correct to write in this manner but to explain the idea I am writing this thing so suppose if I have to find the value of this one by X when X is approaching to zero minus so this means you have to take it as the Procol of this number so if you take the reciprocal then you will get a very large number but with negative sign okay because zero minus is a negative number so 1 by zero minus this will be nothing but minus infinity so left hand limit is minus infinity now comes the right hand limit so right hand limit will be limit X tending to 0 plus 1 by X so now this will be 1 by 0 plus 1 by 0 plus means now you have to take the reciprocal of this number is you know plus so if you take a reciprocal of a number close to 0 that you will get a very large positive very large number and since this is positive number it will be a large positive number so 1 by 0 minus is minus infinity 1 by 0 plus this will be plus infinity ok so again since left hand limit is minus infinity right hand limit is plus infinity left hand limit is not equal to right hand limit this means limit does not exist ok now if you see if you want to see this thing graphically then we all know how to draw the graph of this function y equals to 1 by X this y equals to 1 by X is nothing but rectangular hyperbola ok this we have seen in our earlier discussion when we were talking about symmetry about the line y equals 2x so this was our graph of the function y equals to 1 by X so here you can see that we had to evaluate the limit at x equals to 0 so if you are approaching to 0 from left side the value of Y is approaching to minus infinity right when X is very close to 0 Y is minor infinity so this means your left-hand limit was minus infinity and when you are approaching to zero from right side the value of y is approaching to plus infinity right so on one side you are getting minus infinity from other side you are getting plus infinity and therefore here limit does not exist okay because the left-hand limit and right-hand limit are not the same so this was our second example now we are going to see our third example and from these two examples second and third we will derive one important result our third example was this limit X attending to 0 1 by X square okay now we need to evaluate now we need to find whether this limit exists or not and if it exists then what is the value of limit so here if I want to calculate left-hand limit then left-hand limit will be left hand limit will be limit X tending to 0 minus 1 by X square right so now this will be 1 by and if I place x equals to 0 minus then I will get to 0 minus a squared ok now 0 minus is a negative number this was our zero minus but if you acquire this then you will get a positive number right and if number is very close to 0 then a square is also close to 0 so now your denominator is a number is a positive number very close to 0 and therefore se Procol will be plus infinity okay so left-hand limit is plus infinity now we can talk about this right hand limit so right hand limit will be limit X tending to 0 plus 1 by X s square so this will give us 1 by 0 plus s squared now 0 plus positive number square a positive number but a very small number very close to you received we'll be infinity okay so now as you can see that this left-hand limit is equal to right hand limit and therefore limit exists and what is the value of this limit so we can write that this limit extending to 0 1 by X square this will be infinity the reason is that whether you are approaching to 0 from left side or right side you are getting the same value plus infinity so therefore here limit exists and if you want to see this thing graphically then if these were our coordinate axis then this was our graph of function y equals 2 who won by X s square right this was our graph of y equals to 1 by X s square this graph we had seen in one problem unbounded and unbounded function so if you see carefully here if I we have to evaluate the limit at x equals to 0 at 0 the function is not defined when we are approaching to 0 from left side or from right side the value of y approaching to same number plus infinity right left hand limit is also plus infinity the right hand limit is also plus infinity so therefore here we can say that this limit exists and the value of limit will be plus infinity now I have taken these two problems to explain the concept of limit because many times people have this confusion that if I am getting plus or minus infinity limit then there must be something wrong I mean you must get some nice number like 2 3 4 5 or some finite number in case of limit but that is not correct here you can see that in first example here limited does not exist but the reason is not that we are getting plus or minus infinity the reason is that we are getting different value for left hand and right hand limit so therefore here limit does not exist but in the second example you can see that you are getting plus infinity and plus infinity for both left hand and right hand limit and therefore here limit exists okay the value of limit is infinity is perfectly valid answer only thing is that you should get either plus infinity or minus infinity so for both left-hand and right-hand limit if you are getting different value for left-hand and right-hand limit then limit will not exist so this was all about existence of limit now when I talk about indeterminate forms