0:00 today we will study one of the most important topics in calculus and that is limits the first question that comes to mind is why do we study limits Albert Einstein famously said once we accept our limits we go beyond them limits help us explore the boundaries of the known and unknown and thus the other embodiments of the human Spirit of Relentless inquiry in calculus as well as in life some functions vary continuously While others jump or vary erratically to investigate the behavior of such functions and analyze them we need the concept of
0:45 limits now suppose we want to pursue research in something of which nothing is known previously then how do we approach it a logical way to move forward is to approach the unknown in its neighborhood and accumulate more and more information by moving closer and closer in the same way in calculus we will study many cases where a function may not be defined at a point say for example this case when the function value is missing at x equal to a here we know nothing about the value of f of a so what we do is explore the function in the neighborhood of a and get as close as possible to a from both sides the
1:33 left hand side and the right hand side along with the missing values and Boundary problems we will be studying the slope of chords in a curve let us assume a function y is equal to f of x and a b is a chord of the curve ephedral lines parallel to the axis and make a right angle triangle with an angle say Theta then tan theta equals Delta y upon Delta X now we want this B to be as close as possible to a so that the chord becomes a tangent at a for this to happen B should be nearly a but not a in this case we can take a limit of B
2:19 approaching a and we get derivatives and differentiation another fundamental problem in calculus is the area under the Curve suppose we wish to calculate the area under the curve for this function y is equal to f x between X is equal to a and X is equal to B hey what we will do is we will divide the area into n equal parts and draw rectangles from the graph we can see that the rectangles overshoot or undershoot the curve to calculate the area of the curve we will have to divide the region into Infinite parts now Infinity is not a number but an idea of something unlimited endless and without Bound in
3:07 this case we can take a limit of n approaching infinity and we get integration Without Limits there will be no calculus for there will be no differentiation or integration it will then just be the study of functions through trigonometry and coordinate geometry so in the end I would just say know your limits but never stop trying to exceed them now before we study limits there are some basic terms that we must know and the first thing is neighborhood of a point
3:53 if on a real line we take a point say x equal to a we can Define neighborhood of a bar this symbol n a and it represents all the points or all the real numbers in a symmetrical open interval around this point a so if we take positive number say Delta such that this length is Delta then this point is a minus Theta and this point is a plus Delta and this interval a minus Delta to a plus Delta it is
4:39 called as neighborhood of a now this Delta it is called as radius of neighborhood so basically neighborhood of a point it is a set of all values of x which lie in this interval from a minus Delta 2 A Plus Delta and we can also write this as x minus a it lies between minus Delta n plus Delta and another way to write it is this mod x minus a it is less than Delta
5:25 and when we specify Delta we also write this neighborhood as n a Delta sometimes we talk about deleted neighborhood of a now when we say deleted neighborhood of a that means this point a it is missing from this neighborhood we'll write this as this a minus Delta 2. a plus Delta minus this point a then this deleted neighborhood if we look at this interval that is from a minus Delta to a we call this
6:13 left hand neighborhood of a and then points on the right from a to a plus Delta and the set containing all the points on right of a that is from a to a plus Delta it is called as right hand number right hand neighborhood of a so if we say there's a point say x equals to name this point is x equal to 2 and if we Define a very very small Delta then this interval from 2 minus Delta
6:58 to 2 plus Delta it is called as neighborhood of 2 and we'll write this as n 2 Delta and all the irrational rational points in this interval they are included in this neighborhood so basically we are talking about all X that lie between 2 minus Delta 2 plus Delta or we can write this as this mod x minus 2 and it is less than some positive very small Delta now the second thing that we need to learn in limits is the concept of
7:45 X tends to a so when we write this x Arrow a it simply means that this X it approaches a or X tends to it now when we say x approaches a or X tends to a it simply means that X is nearly a but not a that means we should keep in mind when we write X tends to a X's
8:33 unequal to a so if we write X tends to 2 then it means X is unequal to 2 but X is nearly 2 and Loosely speaking this x can be 1.9999 and this 9 after a long time or it can be this x which is 2.000 and this value 1. if we look at this in terms of neighborhood of a then this point is x equal to a and then we have this neighborhood of a from a minus Delta to a plus Delta
9:22 and if we say X approaches a that means we are moving through values of x in this interval on the left towards this point a or we are moving through values on right towards this point a so we'll move through left neighborhood and right neighborhood towards this point a and we'll get as close as possible to a so that it is nearly a but still it is not a and since we can approach this point from two different directions from left hand side and from right hand side of a
10:08 we can further Define the symbol extends to it if we are approaching this a from the left we say this is X tends to a negative and we are approaching this point from the right then we say it is X tends to a positive now this is not minus a or a to the power minus 1 it is just a symbol which represents we are approaching this point a from the left hand neighborhood of a and this symbol extends to a positive it means we are moving towards a from the right neighborhood of a so this point one point nine nine nine nine nine it is in the left neighbor of this point too and this point two point zero zero zero zero zero one it is in
10:54 a right neighborhood of a so if we write X approaches 2 negative that means Loosely speaking we can say we are moving from 1.9 to 1.99 to 1.999 to 1.99999 and we keep getting as close to 2 as possible and mind it I said Loosely speaking because in the neighborhood it is not just the rational points it'll have rational as well as irrational points in the same way Loosely speaking X approaching to positive then X can
11:40 take the value say 2.1 and then we'll move closer to 2.01 and still closer to point zero zero one two point zero zero zero one and we can keep approaching to from the right hand side so this is the symbol X approaches a and with this we can write this limit notation so if we have to write limit of a function f x at a point a we write this limit as
12:25 this symbol align for limit and extends to a this function f x and it is equal to some value say l and this is how we write limit of the function and when we say this limit X tends to a f x equals L it simply means that value of this function f x it approaches this value l when X approaches a and clearly in this axis are equal to a and f x is unequal to L it is just that when X is
13:11 nearly a f x is nearly L and this is what limit notation means with this we can now Define informal definition of a limit [Music] if we have a function f x which is defined in the deleted neighborhood of a that is it is defined in the neighborhood of a except possibly at x equal to a if f x goes arbitrary close to L for all X in the
13:58 neighborhood of a that is whenever X it gets arbitrary close to a then we write this limit extends to a f x equals l so here we have not defined limits formally but rather intuitively now if we look at it graphically then suppose these are our X's we have this function f x this is the point
14:45 x equal to it and this is the value l so in this informal intuitive definition when X approaches a either from the left or from the right this function f x it approaches this point from the left and also from the right so value of this function FX it approaches l from both sides and this is the intuitive definition of limits now we'll discuss formal definition of limit or Epsilon Delta definition of limits
15:33 now suppose we have this function f x which is defined in an open interval containing a except possible yet x equal to a that is this function f x is defined in deleted neighborhood of a and let L get a real number then we'll say this limit
16:18 X tends to a f x it is equal to l if for every Epsilon greater than zero there exists a Delta greater than zero such that mode of x minus a it is greater than 0 but less than Delta then mod of f x minus L is less than Epsilon now what does this mean it simply means that the limit of this function f x is L if for any Epsilon which is greater than 0 we can Define
17:03 a neighborhood for this value l so this point it is L minus epsil and this is L plus capsule so we have this neighborhood of L for every Epsilon greater than 0 it must have a Delta which is radius of deleted number of this point a so this is a minus Delta and this is a plus Delta and this point a it is deleted from this neighborhood so basically former definition is whenever we choose any Epsilon can we Define neighborhood of this point L corresponding to that neighborhood we must get a neighborhood
17:48 of a and whenever we get this condition we'll say limit of this function is equal to l so hover small this Epsilon may be if we get a corresponding Delta every time then we'll say limit of the function axis and this limit extends to a of X it is equal to n now let us take an example say for example using formal definition we have to prove that this limit extends to 2 x minus 1 it is equal to 3. now in this case a is 2 L is 3
18:36 and F axis 2x minus 1. now what we'll do is we'll say we have some Epsilon greater than 0 then f x minus l it's mod lb less than Epsilon now f x is 2x minus 1 and then L is 3 it is less than Epsilon now this is mod of 2x minus 4 and this less than Epsilon we can take this two Commons will be 2 into mod of x minus 2 it is less than Epsilon or we can write mod of x minus 2 it is less than
19:22 Epsilon by 2. now if we Define Delta s Epsilon by 2 we can write mod of x minus 2 it is less than Delta so for every Epsilon greater than 0 we'll get a Delta greater than 0 such that x minus a which is mod of x minus 2 is less than Delta so it means limit of this function when X approaches 2 is 3. now once we have defined limits we'll learn how to solve limits but before that we need to learn something which is fundamentally very important in limits and which is indeterminate forms
20:19 now whenever we have to solve question and limits say limit X tends to a f x what we'll do is we'll try and find value of this function by direct substitution and when we try to find this value by direct substitution you'll get certain forms in which through the limiting behavior of individual parts will not be able to determine the overall limit so these forms are called as indeterminate forms now typically there are seven important indeterminate forms and they are 0.0 form
21:05 Infinity upon Infinity form Infinity minus infinity form 0 into Infinity form 1 to the power infinite form 0 to the power zero form and Infinity to the power 0 forms so these are the seven most important indeterminate forms that will come across in the limit questions so whenever we are solving any limits question the first thing that we'll do is we'll put the value of this a in this function and by direct substitution we'll find whether or not we have an indeterminate form take for example this case this limit
21:53 extends to zero sine X upon X now in this case if we put X as 0 sine 0 is 0 and x 0 is 0 so this is 0.0 form which is one of the indeterminate forms and if we take this another case which is limit extends to Pi by 2 sine X upon X now in this case sine Pi by 2 is 1 and X is pi by 2 so this value is 2 upon Pi which is not n [Music] indeterminate form
22:39 so identifying the form it is one of the basic steps in limit and we'll be doing it for all the levels in this video we'll be discussing why these seven standard forms and limits are called as indeterminate forms we'll begin with 0 upon zero now in algebra we know that division by 0 is not defined so we can say a upon B will be defined if and only if B is unequal to zero and if D is unequal to 0 then this a upon B will be some value l which we can also write as a equals L into B now in calculus also division by
23:24 0 is not defined now for this function f x upon J X to be defined we have to exclude values of X at which denominator is 0. so in calculus also division by 0 is not defined except in the case of limits so if we have any limit say limit extends to a f x upon GX where this limit extends to a f x it is 0. and limit X tends to a GX it is 0.
24:11 now it becomes 0 upon zero form now the question that naturally comes to the mind is y 0 upon 0 defined in case of limits the answer to this question lies in the concept of limits when we say limit X tends to a f x it is zero it simply means that when X approaches a value of f x it approaches zero f a zero but f x when X approaches a is not 0 it approaches 0 so that means value of f x is not zero it's a very small number close to zero but still it is not 0 in the same way when X tends to a g x also
24:58 it approaches 0 it is not exactly zero so here when we say it is 0 upon 0 form it is nearly 0 upon nearly zero so here also it is not division by zero it is division of very small number where another very small number now when X approaches a GX approaches 0 g x is not equal to 0 and when GX is not 0 then in that case when B is n equal to 0 we can define a upon B and we can write a upon b as sub number l so now we can write limit extends to a FX it is L and limit
25:45 X tends to into GX now this value on the left hand side it is nearly zero and on the right hand side also this is nearly 0 0 multiplied by anything is 0 so this form 0.0 in limits it may take any value and that is the reason we call this form as indeterminate form now say for example suppose we have this limit which is limit X tends to zero ax upon BX now it will put X as 0 then it is 0.0 form now since X approaches 0 X is not exactly 0 we can
26:30 cancel this x with x then the answer to this limit will be a upon B now choosing different values of a upon B we'll get different answers for 0 upon zero suppose we take a as 1 and B S2 so then we can write limit X tends to 0 x upon 2X it is 0.0 now this value is 1 by 2. if we take a S2 and B as 1 we'll get 2x upon X which is 0.0 this is 2. now another example could be limit X tends to zero ax upon X Cube now this again is 0.0 form now we can cancel x with X it'll become
27:18 limit extends to zero a upon x square now this value lb plus infinite when a is greater than 0 and it'll be minus infinite when a is less than 0 so this 0.0 form it may take any finite value it may take values plus infinite and minus infinite and also if we consider this case limit X tends to 0 x square upon greatest integer function of x square now this again is 0 upon zero form but in this case when X approaches 0 x square approaches 0
28:05 now greatest India function of x square it doesn't approach 0 because greatest unit function can only take integral values it cannot take approaching values so in this case greatest need a function of x square is exactly 0 and not limiting value zeros in this case this limit it won't be defined another example that we can consider is this limit limit X tends to 0 X cos 1 upon X upon 2X now this is again 0.0 form now again we can cancel x with X it'll become 1 by 2 cos infinite and cos
28:52 infinite is oxalating value between minus 1 and plus one so the value of this limit will oscillate between minus 1 by 2 N plus 1 by 2 and since there is no finite single value this limit does not exist so in all these cases we have seen that this form 0 upon 0 it may not be defined it may take Value Plus infinite minus infinite it may be zero it may take any finite value and since we do not have any predetermined value or any predetermined result for 0 upon zero we call it as indeterminate form now this second and third forms they correspond to First from 0.0 so if you have any limit which is limit
29:39 X tends to a f x upon g x which is of the form Infinity upon Infinity which simply means this limit X tends to a f x it approaches infinite and also this limit X tends to a GX it approaches infinite we can rewrite this limit as limit X tends to a 1 upon GX upon 1 upon FX which will make it 0.0 form and 0.0 form we know is indeterminate so answer to this limit again will be indeterminate so this Infinity upon Infinity it must also be an indeterminate form also if we look at
30:25 this limit which is 0 into Infinite say for example suppose we have limit X tends to a f x into g x which is of the form 0 into Infinite that is limiting value of this function f x is 0. and limiting value of this function GX it is infinite so now it becomes 0 into Infinite form we can rewrite this equation as limit extends to a f x upon 1 upon g x now limiting value of f x is 0 and limiting value of 1 upon g x is again 0 so which will again make it 0.0 form which we know is indeterminate so from here we can say this 0 into
31:11 Infinite will also be in indeterminate form now we'll come to this fourth form which is infinity minus infinity now Infinity minus infinity is not 0. suppose we have a very large number which is infinite now if we'll add one what is infinity plus one infinity plus one is infinity if we will add to infinity plus 2 is again infinite infinity plus L will again be infinite so Infinity minus infinity it can take any value l so again this Infinity minus infinity it may take any form or any value and that is the reason it is called as indeterminate form now this simple result that Infinity minus infinity it is indeterminate can be understood by a thought experiment suggested by
31:56 Hilbert and which goes by the name Hilbert's Paradox of Grand hotel now the Paradox is suppose there is a hypothetical hotel with countably infinite number of rooms all of which are occupied so we have countably infinite rooms one two three four and it goes all the way up to infinite and all these rooms they are occupied now since all the rooms are occupied it is logical to think that hotel would not be able to accommodate any newly arriving guests as would be the case with finite number of rooms now suppose a new guest arrives at the hotel now to accommodate this guest what manager has done is manager has asked every guest to change their room number from n to n
32:41 plus one so one goes to two two goes to 3 3 goes to four four goes to five now this room one is empty now we can accommodate this one in this first room now suppose we have L you guys now in this case manager will ask every person to go from room number n to room number n plus L now we'll have first albums vacant we can fill these L gas in L rooms in the first case we have Infinity is same as infinity plus one in the second case we have Infinity equals infinity plus n if you have infrared guest then what manager will do is manager will ask every person to move from room n to room to n now once he does that all the odd number rooms which
33:27 are countably infinite are vacant where we can accommodate these infinite gas so in this case infinity plus infinity it approaches Infinity so this thought experiment it clearly illustrates this counterintuitive indeterminate property of infinity minus infinity form now we'll come to the next one which is 1 to the power infinite suppose we have any function which is limit extends to a f x to the power GX where this limiting value of f x at a it approaches 1. and limiting value of g x it approaches infinite which makes this form 1 to the power infinite form
34:14 now here this form has to be approaching 1 to approaching infinite it cannot be 1 to the power approaching infinite so as a counter example if we take this limit 1 to the power n now this 1 to the power n is not 1 to the power infinite form it is not approaching one in this case it is simply one so when we have approaching 1 to approaching infinite then we have this indeterminate Form 1 to the power infinite now for all the remaining three forms what we'll do is we'll simply use the fundamental that a to the power B is e to the power so for this first one we can write this
34:59 limit as e to the power limit X tends to a Lon f x to the power g x and lb limit X tends to a g x into Lon f x now in this case g x approaches infinite and f x approaches one so this is e to the power infinite into zero form and we have already established that infinite into zero is indeterminate form so this result will be e to the power l so this form again 1 to the power infinite will be an indeterminate form now for 0 to the power 0 form f x approaches 0 and GX approaches 0 then in this case
35:45 it'll be e and then GX approaches 0 0 and f x approaches 0 is long zero and long 0 is minus infinity so it'll be again e to the power 0 into Infinite which again is indeterminate form and finally when we have this Infinity to the power 0 f x approaches Infinity and g x approaches 0 so then we can write e to the power now g x it approaches 0 and log f x Now log infinity is infinity so this again is 0 into Infinity form which is an indeterminate form so this again is in indeterminate form result of limits of these forms is not predetermined we call these forms as indeterminate forms and we have to solve the question to find out its exact answer and besides these
36:33 we need to know some standard results we can get these results from properties of functions or we can get these results from graphs now if we look at the graph of y equals log it's base a we know that depending on the value of a we'll have 2 graphs now if a lies between 0 and 1 that is the graph
37:22 and if a is greater than 1 we get this other graph now if I am going to ask you this limit X tends to 0 positive log X base a now in this case we want to determine behavior of this function log X when X approaches 0. so in this first case when a lies between 0 and 1 as X approaches 0 so s x approaches 0 y approaches plus infinite so in this case this limit
38:08 extends to 0 positive log X base a it is plus infinite but in the second case when a is greater than 1 then X approaches 0 value of y it is moving towards minus infinite so in this case this limit extends to 0 positive log X base a it is minus infinite now the same way if I am going to ask you for this first case this limit X tends to infinite
38:57 log X base a now in this case when X approaches infinite this one it approaches minus F right so this limit it is minus infinite and in this case when X approaches infinite this y also approaches inference so this is this limit extends to infrared log X base a and l b equal to plus infrared in the same way if we look at y equals a to the power x when a is greater than 0 again we have two graphs this first graph when a lies between 0 and 1 and the
39:44 second graph when a is greater than 1. now this first graph it is a decreasing function and the second graph it is an increasing function this is 0 and this is 1. now we'll be writing these limits limit extends to minus infinite e to the power x and limit X tends to plus infinite a to the power x and here also we'll be writing limit extends to minus infinite a to the power x and limit
40:29 stands to plus infinite a to the power x now here when X approaches minus infinite this is why it approaches plus infinite so this limit it is equal to plus infinite and in this case when X approaches plus infinite this graph it approaches 0. so value of this limit X tends to infinite e to the power x will be 0. now in this case when X approaches minus infinite this graph it approaches 0 and when X approaches plus infinite this graph approaches Plus so we have this important result if a lies between 0 and 1 then e to the
41:14 power minus infinite it is infinite and e to the power infinite is 0 and if is greater than 1 then a to the power minus infinite is 0 and e to the power plus infinite is infinite then in some questions we'll come across this condition limit X tends to infinite sine X now whenever we say sine X or sine infinite now we know that value of sine X will always lie between minus 1 and plus 1 so this sine infinite it doesn't approach his infrared in fact this sine infinite it will be an
42:02 oscillating value between minus 1 and plus 1 so this limit at infinite it will take all the values between minus 1 and plus 1 and the same way if we have to write cos infinite cos infinite again will be this oscillating value from minus 1 to plus one so while solving limits we'll be coming across these notations of sine infinite and cos infinite and we'll be using certain properties for greatest integer and fraction part function so if this box X it represents greatest linear function and N is some integer and H is a very small positive number which is nearly 0 then greatest singular function of
42:48 n minus H it will be n minus 1 and greatest figure function of n plus h it will be simply n say for example if we take this limit 2 minus H now H is very small number and 2 minus H will be 1.999 something so this value its greater function will be 1. write this as suppose 2 plus h now this is 2.000 something this is nothing but 2 but if it is non-integral value say y minus H now this is 3.14 minus h 5 plus h that will be 3.14 plus h it is
43:36 still three point something 3 and if we have this fractional function then for any n which is an integer MH which is 0 positive we know that fractional part of n plus h it will be fraction part of H and we know that since H lies between 0 and 1 fraction part of X is simply X so it'll be this h and if we have fractional part of n minus h it will be simply minus h and we know that fraction part of minus X is 1 minus fraction part of X when X
44:24 is not an integer so L be this one minus a so we'll be using these standard results and we'll work out some more standard results through graphs while working on the problems [Music]
