[Music] so [Music] [Applause] [Music] hello friends welcome to lecture series on multi variable calculus in the last lecture we have seen that what do you mean by limits for several variable functions we have seen that if we write limit x y tending to x naught y naught f x y is equals to l this means if this this if this limit exists and is equal to l it means for every epsilon greater than 0 there exist a corresponding delta greater than zero such that such that mod of f x y minus l is less than epsilon whenever whenever ah zero less than under root x minus x naught whole square plus y minus y naught whole square is less than delta so we have seen that whatever epsilon may be no matter how small how large it may be there will always exist a corresponding delta greater than zero such that this inequality hold that means for every epsilon for every epsilon there will exist a delta such that a disk centered at x naught y naught of radius delta all those x y lying in that disk will always the image of all those x lying in that disk will be contained in l minus epsilon 2 l plus epsilon and that is the geometric interpretation of this definition now let us discuss some important properties of limits the first property is limit x y 10 into x naught y naught f x y if exist is always unique ok next is to find out the value of the limit the another method is convert cartesian coordinate into polar coordinate system that means if we substitute x minus x naught equal to r cos theta y minus y naught equal to r sin theta where r square is equal to x minus x naught whole square plus y minus y naught whole square and tan theta is equal to y minus y naught upon x minus x naught that can easily be obtained if you divide the ah second expression y minus y naught ah is equal to r cos theta and x minus system to cosine theta then we obtain ten theta equal is equal to y minus y naught upon x minus x naught the definition of the limit can be expressed in this way so basically if we are having limit x y tending to x naught y naught which we have just explained you f x y equal to l then to find out this limit the another method is convert this cartesian coordinate into polar coordinate system so how can we do that ah you simply take x minus x naught as r cos theta y minus y naught as r sin theta where ah if you square an add it is simply r square is equal to x minus x naught whole square plus y minus y naught whole square and tan theta is equal to y minus y naught upon x minus x naught ok now as r ten into zero whatever theta may be x will tend to x naught and y will tend to y naught that is x y ten ten x y will tend to x naught y naught ok so that means these are there are two ways either you convert x by ten to zero x naught y naught two x y tend to zero zero and then you can convert this into polar coordinate system or the other way out is you simply take x minus x naught as r cos theta and y minus y naught as r sine theta ok so now now this limit will convert to this will convert to limit r ten into zero because as r tend to zero x will tend to x naught y will tend to y naught that is x y will tend to x naught y naught ok this will be f of r cos theta r sin theta and the limit will be c m l so how we can define this in delta epsilon now so to show there is the existence of this limit again we will use the concept of delta epsilon that is for every epsilon greater than zero there will exist a corresponding delta greater than 0 such that it is mod r less than delta implies mod of r cos theta r sin theta minus l less than epsilon for all theta for our r and theta this must hold for all for all r and theta ok so so so any cartesian coordinate ah if you have a limit to find out in any cartesian so either you can ah either you can proceed in the cartesian way only or you can convert this into polar coordinate system to find out the limit now for example we have this problem ah changing into polar coordinate system show that limit of this is equal to zero now let us try this problem so we were discussing about that how we can show ah existence of a limit by delta epsilon definition so let us consider this example that is limit x y tending to zero zero x cube upon x square plus y square the limit is zero ok now let us try to prove this that this limit is zero so we have two ways to show this the first way is convert x and y into polar coordinate system ok and the second way is you can proceed by the ah usual cartesian method ok so let us first try to prove it by like converting this into polar coordinate system so we will suppose that x equal to r cos theta and y is equal to r set theta now as x y both are tending to zero zero so it will be possible only when r will tend to zero ok so this limit will convert into limit r ten into zero x is r cos theta so it is r cube cos cube theta now r square cos square plus r square sine square is r square because cos square plus n square theta is one so this is equals to limit r tending to zero it is r cos cube theta and this is clearly zero for any theta you can if you take any theta the if limit r is tending to zero this will always tend to zero now to show this that this is equal to zero we will again use delta epsilon definition ok so let epsilon get on zero be given ok so it is mod r cos cube theta minus zero which is equals to mod r cos cube theta which is equals to mod r into mod cos cube theta which is less than equals to mod r into one because mod cos theta is always less than equal to one so and this is less than delta so if we choose if we choose delta less than equals to epsilon then mod of r cos cube theta minus zero will be less than epsilon whenever zero less than mod r less than delta so hence we have shown the existence of such delta for which this inequality hold hence we can say that this limit exists and is equal to zero ok now the same can also be proved by by the by the usual like delta epsilon definition without converting this into polar coordinate if you want to prove this result without using a polar coordinate then also we can do that without using polar coordinate also we can prove this limit let delta f let epsilon greater than zero be given we have to show that ah mod of x cube upon x square plus y square minus zero is less than epsilon whenever zero ah less than under root x minus zero whole square plus y minus zero whole square is less than delta this we have to prove we have to prove the existence of such delta ok now we take this inequality mod x cube upon x square plus y square minus zero this is equals to mod of x into x square upon x square plus y square this is further equals to mod of x into mod of x square upon x square plus y square now x square is always less than equals to x square plus y square ok so x square upon x square y square is always less than equal to one and it is also non negative quantity so we can say that it is less than equals to mod x into one ok and this mod x now you can use the other definition of you can use the other definition of limit or mod x cube upon x square plus y square or if you want to use same you can do same also so we can use this definition here so if you take this is less than delta so choose delta equal to epsilon then we have done so we can prove this ah existence of this limit without using polar also but if you use polar coordinate system then we can get the result easily okay ah other properties of limit limit is two path test for the non existence of a limit a from two different paths as as x y approaches two x naught y naught the function f x y has different limits then this implies limit does not exist ok so what does it mean let us see now we are having x axis y axis we have a point x naught y naught ok this point is basically x naught y naught we take a neighborhood of this point this point x naught y naught ok now if you take any neighborhood of x naught y naught ok all those x y lying in this region there are infinite paths by which this x y can approach to x naught y naught it may be a straight line it may be a parabola curve it may be some other curve ok there may be infinite paths ok now existence of limit means if we follow any path from x y to x naught y naught it must be path independent path independent means whatever path we follow from x y to x naught y naught the value of the limit will always be unique if the if the value of the limit if you if you are saying that limit x y tending to x naught y naught f x y is l ok this means this means if we take a neighborhood of x naught y naught and we are taking any x in this neighborhood they are infinite paths from by which x y can approach to x naught y naught it must be path independent that means whatever path we follow from x y to x naught y naught the value of this limit is always l will always remain the same because it is because limit is unique limit is always unique if it exists that means if from two different paths value of the limit are not same value is a double limit we are calling it double limit if the value of double limit are not same this means limit does not exist because if limit exist this means it must be path independent so to illustrate this let us discuss few examples ok the first example is limit x y tending to x naught y naught x y upon x square plus y s square now ah ok zero zero it is zero zero now from x y to zero zero they are infinite paths ok we can follow any part suppose this is origin ok and this is any x y this is any x y so we can we can move along x axis we can move along y axis we can move along y equal to x we can move along y equal to two x we can move along y equal to x square the infinite paths ok so let us move along let us move along x axis or y equal to zero now if we move along y equal to zero if we move along y equal to 0 from this point to this point from this path we are following if you follow this path then what is the limit of this expression there will limit x into 0 y equal to 0 okay and x y upon x square plus y square and it is when you substitute x equal to zero when i substitute y equal to zero so the value is zero ok now now let us move along along say y axis or x equal to zero now if you move along x equal to zero it is little bit y tending to zero x y upon x square plus y square and x equal to zero when you substitute x equal to zero here this is zero now from these two paths value are same what does it mean from if if from two part two different paths value of the double limit are same it means the double limit that is this limit may or may not exist because these are only two paths and they are infinite paths from x y to zero zero they are in finite paths ok and if from two pass value are same it does not mean that the value exists when your double limit exists and is equal to zero because there may be some other path from which the value of this limit may be different for example if you take say y equal to x if you take if you move along y equal to x along y equal to x then limit x into zero you substitute y equal to x it is x square upon x square plus x square which is limit x into zero x square upon two x square which is one by two now from this path from this path value is zero from this path value is zero and from some other path value is one by two so values are not same values are different this means this limit does not exist so we can simply say this implies limit x y tending to zero zero x y upon x square plus y square does not exist ok y does not exist because from two different paths values are different ok now the other way out to show that ah limit does not exist is other way out is you take you take a path general path ok you take a general path you move along say y equals to m x if you move along y equal to m x this means it is limit you substitute y equal to m x it is x into m x upon x square plus m x whole square and x is tending to zero remember this that this path must pass through x naught y naught must pass here x naught y naught is zero zero so this path must pass through as zero zero ok whatever path we are choosing it must must pass through this point now this is equals to limit x into zero it is m x square upon x square times one plus m square x square cancel out and it is m upon one plus m square now this value the limit this value comes out to be dependent on m you take different values of m say you take m equal to one this value is one by two you take m equal to two then this value is two upon five so for different values of m the value of the limit are different this means limit does not exist because now it is path dependent we take different paths values are different it is part dependent however it must if limit exist it must be path independent so it depends on m this implies limit x y tend to zero zero x y upon x square plus y square does not exist so basically to show to show that limit does not exist the double limit does not exist we have two ways the first way is you take two different paths and try to show that from two different paths value of the limit are different the other way out is you try to show that it is path dependent you take some arbitrary path like y equal to m x or y equal to k x square or something and try to show that it is it depends on m or k okay in this way we can show that limit does not exist say we have second example it is limit x y ten into zero zero the problem is x cube y upon ok now if you move along if you move along say y equal to zero if you move along y equal to zero then this value when you substitute y equal to zero then this is clearly zero ok because when you substitute y equal to zero here this is zero now you move along say y equal to x cube along this curve if you move along y equal to x cube then this is nothing but limit x ten into zero x cube into x cube upon x k power six plus it is x to the power 6 which is equal to limit x 10 to 0 x to the power 6 upon which is 1 by 2. so from one path value is zero and from other path value is one by two this means this limit does not exist now the next example next example is limit x y z tending to zero zero zero it is x y z x square plus it is y to the power four plus z to the power four [Applause] now we have to find a path such that it comes out to be path dependent to show that this limit does not exist ok so we can choose some path say we can take let x is equals to some k t square say y equal to y equal to say t and z equal to t where t is some parameter ok basically in ah in three d we are taking this ah curve okay now when you substitute this it is limit x is k t square y is t and z is t and it is k square t raise to power 4 plus t raised to the power 4 plus t raised to the power 4 and limit t ten to zero because because as x y z all are ten to zero this will happen only when t with t tending to zero ok and this is equals to limit t ten into zero it is k into t raised to power four upon upon ah k plus k square plus two so this will be equal to k of k upon k square plus 2 that is depends on k it depends on k this means this means this limit does not exist ok so in this way we can show that double limit does not exist now if you take say if you take four five paths and the value of the limit always come out to be same then again this does not guarantee that the limit exists because there may be some some other path by which the value of the limit comes out to be different if we have to show the existence of a limit we have only option is delta epsilon definition we have to show the existence of a limit using delta epsilon definition only to show that the limit does not exist we can use this this concept we can we can use two different path and try to show the value of limit comes out to be different or where we can try to show that it is path dependent now we will talk about iterated limits and double limit now what does it mean you see that double emitter is this thing this is x y and into x naught y naught f x y suppose it exists and equal to l ok and we have it is called double limit also called double limit ok and other things are iterated limit it limit means limit x tend to x naught limit y time to y naught f x y r limit y tend to y naught limit x into x naught f x y these are called iterated limits ok now now if you take now if you take x naught y naught here ok and you take a neighborhood of this point center at x naught y naught you take any x y in this disk this means you first you first take y ten to y naught keeping x constant and then you take x x ten to x naught so first you are taking y tend to y naught this is ah this is this is x naught y naught ok first you are taking y tend to y naught means this thing ok y tending to y naught ok this is some point x y ok y tend to y naught now this now this point is ah this point is x naught y naught ok now here first x n two x naught and then y tend to y naught so we come to this point ok so this is y tend to y naught and then x n two x naught so we come to this point ok ok so t these are these are basically two different paths one path is this another path is this ok now ah now if this limit exist and is equal to l then the iterated limit then the iterated limit value of the iterated limit is also equal to l provided provided limit y ten into y naught f x y and ah limit action x naught f x y exist if this condition hold then only we can say that ah if double limit exist then iterated limit also exist n equal to l so basically if this is equal to l then this implies then this condition implies if this is equal to l then this condition implied that this these are also equal and is equal to l provided provided this inside limit exists because because if this limit exist then these are basically two paths ok if this limit exists then these are basically two paths and if this is equal to l this means it is path independent if if it is path independent then from these two paths also the value will be same value will be l now if you see the converse part if this is if this exists and suppose this and this are equal to l then then these are two only two parts ok if this and if this and this limit exist then these are only two paths and from these two paths if limit comes out to be l then this double limit may or may not exist because basically these iterated limit if this limit exist are only two paths ok so let us understand ah this by giving some examples you see suppose you take this it is x plus y upon x minus y suppose you want to compute this this limit x minus y should not equal to zero now now if you find this limit limit x into zero limit y tending to zero x plus y upon x minus y if you find this limit this is this iterated limit then this is limit x into zero you simply substitute you simply tend y tend to zero then it is x plus zero upon x minus zero and when you take action to zero then this is one ok now you take the other iterated limit it is limit y tend to zero it is zero plus y upon zero minus y and when you take y ten to zero it is minus one so iterated limit exist and are not equal ok you see you see that this this limit and this limit exist this limit is one and this limit is minus one this and this limit exist ok and it rated limit are not same this means this implies limit x y 10 to 0 0 x plus y upon x minus y this does not exist ok because if because if this inside limit exists then these iterated limit are simply two paths and from the two different path values are different this means this limit double limit does not exist ok now see another example it is limit x y tending to zero zero it is x square y square upon it is again x square y square plus x minus y whole square ok now problem is find a double limit and iterated limit if they exist the provided denominator is not equal to zero now first you find the double limits ok so you take limit y tend into zero limit x n to zero f x y which is x square y square upon x square y square plus x minus y whole square now when you put x when you take x ten to zero here in this expression so this this will tend to zero then this is simply equal to zero ok one can easily see that when you take x ten to zero in this expression so numerator is zero so the value is zero now the other iterated limit is limit action to zero suppose and limit y tend to zero x square y square upon x square y square plus x minus y whole square now when you take y ten to zero again numerator is zero so this value is again zero now these these inside limits exist and the iterated limits are same what does it mean what can we say about double limit from here we can say that double limit may or may not exist because these are only two paths it may be possible from from some other path value or double limit comes out to be different from this limit from this value say say if you take a path ah say you take a path along say take a path y equal to x if we take a path y equal to x here so we will obtain limit x tend to zero x square x square upon x raised to the power four plus zero which is one from this path we are getting value one and from other parts we are getting a value zero this means limit does not exist because there are two different paths from from which value of the limits are different though this means the path dependent then this implies this limit does not exist ok so hence we can easily show that whether a limit exist or it does not exist to show the existence we have to go only through delta epsilon definition to show that the limit does not exist we have to we have to show that from two different paths values of the limit are different ok so thank you very much [Music] [Applause] [Music] you