Limits for scalar-valued multivariable functions

Summary

Highlights

• The lecture revisits limits for sequences of real numbers, introducing concepts of convergence and divergence. 🧮
• Extending these ideas to higher dimensional spaces, the lecture discusses sequence limits in Rp. 📐
• Focus shifts to scalar-valued multivariable functions, generalizing notions from one-variable calculus. 📊
• The discussion emphasizes the importance of checking limits for each coordinate independently. 🗺️
• Examples demonstrate how substitution might fail in complex expressions leading to non-existent limits. ❌

Key Takeaways

• Understanding limits in multivariable functions builds upon concepts from one-variable calculus. 📈
• Convergence and divergence in higher dimensional spaces involve examining each coordinate's limits. 🔍
• Applying limits to scalar-valued multivariable functions often requires techniques beyond simple substitution. 🚫
• The lecture illustrates using examples how different sequences can result in diverging limits. 🔄
• Careful analysis of limits in multivariable contexts can reveal non-existent limits. 🤔

Overview

Limits for scalar-valued multivariable functions Transcription

• 00:00 - 00:30 Hello, and welcome to the Maths 2 component of the online BSc program on Data Science and programming. We have seen multivariable functions and today we are going to discuss Limits for Scalar Valued Multivariable Functions. So, let us recall first that we have seen
• 00:30 - 01:00 the notion of limits for sequences of real numbers. So, if an is a sequence of real numbers, we say that the an has limit a, if as n increases the numbers an come closer and closer to the number a. So, a sequence an is called convergent if it converges to some limit, so that means there is some a, such that an tends to a, an that is an comes closer and closer to a. Here is a couple of examples. So, 1 by n summation 1 by i factorial, n times sin 1 by n. And
• 01:00 - 01:30 in a minute, we will quickly recall what these limits are. A sequence an is called divergent, if it is not convergent. And examples of such sequences are minus 1 to the power n, just n, and then log of 1 by n. So, in a minute, we will see, we will recall what happens to these as well. And also, we defined the notion of a subsequence. So, a subsequence of a sequence is a new sequence,
• 01:30 - 02:00 which we formed by possibly excluding some entries of the original sequence. So, you choose only some of the entries of the original sequence. So now, the goal is to move from real numbers to higher dimensional space, so Rp. And in a minute, we will talk about sequences there and convergence over there. So let us recall first that what happens to these sequences. So, for these sequences, 1 by n, for example,
• 02:00 - 02:30 tends to 0, summation 1 by i factorial, if you remember this one, it tends to e, and then n times sin 1 by n, you can think of this as sin of 1 by n divided by 1 by n. And as n tends to infinity, 1 by n tends to 0. And we know that sin x by x tends to 1. So, this tends to 1. So, this was one of the non-trivial limits that we actually computed
• 02:30 - 03:00 in, back when we did one variable calculus. And then we had, these are divergent minus 1 to the power n is divergent, because it keeps oscillating. So, minus 1 to the power 1 is minus 1, then squared is 1, then again minus 1 1, so it keeps oscillating, it does not come close to any number. So, this just diverges because it oscillates. The number n diverges, but it diverges in a different way, than minus 1 to the power n, because we say diverges to infinity. So,
• 03:00 - 03:30 what happens is, it keeps increasing, and it becomes larger and larger on the positive side, and it goes to infinity, so we say that this diverges to infinity. And then similarly, if you take log of 1 by n, if you recall the log function, then this diverges to minus infinity. So, in one variable calculus, we had these special things called
• 03:30 - 04:00 infinity and minus infinity. And we had these sort of three different notions of divergence, one was just where it does not come close to anything. And the other two were where it diverges, but it diverges to infinity, and the third was where it diverges to minus infinity. So, now, we are going to use this as a basis for exploring what happens in higher dimensional space. So, limits of sequences in Rp. So, let an tilde be a sequence in Rp. Let us recall that
• 04:00 - 04:30 we have been using tilde to emphasise when we are in higher dimension. So an tilde means it is a vector or an element of Rp and it has p components. So, there are p real numbers. So, those coordinates are denoted by an1, an2, anp for each an tilde. So, you have a sequence of such things. And for each one you have p many coordinates. So, you can look at the coordinate sequences in particular. So, for each coordinate, suppose
• 04:30 - 05:00 I decide to fix the first coordinate, so for the first coordinate, I have a11, a21, a31, a41, and so on. So, that is the first coordinate sequence. Then you have a second coordinate sequence, a12, a22, a32, a42 and so on and then all the way up to the pth coordinate sequence. So, that is a1p, a2p, and so on. So, we have such a sequence and, in the sequence, consists of vectors, or elements of Rp. And
• 05:00 - 05:30 each of those have p coordinates, that is the main thing you have to remember. And each of those coordinates forms a sequence in R, in the sense that we studied on the previous slide. So, we say that an has limit a tilde, which is some other vector or element of Rp, where the coordinates are a1, a2, ap if, as n increases, the sequence in the ith coordinate
• 05:30 - 06:00 has limit ai. So, as I said, what you do is, let us say you take the first coordinate, so from this sequence an tilde, for each of these vectors, you extract the first coordinate. So, then you get a11, a21, a31, and so on, that forms a sequence in R and you ask, does it have a limit? Well, if it has a limit, let us say suppose the limit is a1. So, you keep that in store, then you go to the second one, you do a21 sorry a12, a22, a32, a42 and so on.
• 06:00 - 06:30 Does that have a limit? Well, if that has a limit, you call that limit a2, and then you go on. And you do this for all p coordinate sequences. And if all of them have limits, then you put that together into a vector, which we have called a tilde. And then we say that this an which should have come with a tilde, so an tilde. So, we say that an tilde has limit
• 06:30 - 07:00 a tilde, which is this which has coordinates a1, a2, ap, if for each coordinate, the sequence tends to the corresponding coordinate of a tilde. So, ani, the sequence ani tends to ai. So basically, the idea of convergence of sequence says, so limits of sequence says in Rp is just borrowed from R, it is nothing very special, what you do is you look at each coordinate,
• 07:00 - 07:30 ask if the sequence over there converges, and if it does converge, you put them all together, that is all that the limit in Rp does. So, a sequence an tilde in Rp, is called convergent if it converges to some limits. So now, beyond this, the definitions are exactly the same as we have for R. So, a sequence an tilde in Rp is called divergent if it is not convergent. And then we will define something called a subsequence of a sequence, which is given by deleting some
• 07:30 - 08:00 terms, so excluding some entries in that sequence, or you could take all of them or you could delete some of them, exclude some of them. So, this is exactly in parallel with what happens in R. And in fact, to check limits in Rp, you have to check limits in R, that is what this is saying. So here is a couple of examples. So, the first example is 1 by and comma n times sin one 1 by n. The second example is minus 1 to the power n comma n times sin 1 by n. The third
• 08:00 - 08:30 example is a bit bigger, so this is cosine n by n. Some complicated looking expression summation 1 by factorial and then n times cosine of 1 by n. So let us ask what happens to each of these and visualise the first two. So, the first sequence was 1 by n, n times sin of 1 by n. So, let us see how this behaves.
• 08:30 - 09:00 So, I am plotting this over the range of n as n goes from 1 to 500. And we will keep track of this point a. So, when n is 1, this point is 1, 0.84. So, 1, because 1 by 1 is 1, and, 1 times sin of 1 by 1. So, 1 times sin 1. So, this is 1, sin 1, so sin 1 is 0.84.
• 09:00 - 09:30 And as you increase in, you can see what is happening to this point, it started coming closer and closer and closer and closer to the point 0, 1. And this is at 156, it is already quite close. And you can see it is closing in further and further as we increase it. So, if we make this larger, so it is still not reached, but it is coming very close.
• 09:30 - 10:00 And you can see it is moving, moving, moving, moving, and it is coming to 500, where it is almost at 0 comma 1. So, if we make it still larger, you can see it is not exactly at 0,1. But, but yeah, it is very, very, very close. So that says that as n tends to infinity, this sequence, so each coordinate, we know what happens. So, for 1 by n 1 by n tends to 0 as n tends to
• 10:00 - 10:30 infinity. So that is the first coordinate. So, that is why in the first coordinate, we have 0. And in the second coordinate, we have n times sin 1 by n, which we know because sin x by x as x tends to 0 tends to 1, we know that the second coordinate, the sequence tends to 1. So, the limit is 0 comma 1. And we have seen a geometric explanation for that. So, for the second sequence, we have minus 1 to the power n comma n times sin 1 by n.
• 10:30 - 11:00 Now we know from from what we just did that minus 1 to the power n oscillates, and n times sin 1 by n converges to 1. So, this function is, this sequence is also going to oscillate. Let us see the behaviour of of the sequence. So, as it increases, you can see that the point b is changing from, changing value from one side of the axis to the other.
• 11:00 - 11:30 So, as I, as I am at 1, this is minus 1 comma 0.84, then we increase it to 3, it is minus 1 comma 0.98 then we have n is even, say n is 8, this is 1 comma 1. And after that, it just oscillates between minus 1 comma 1 and 1 comma 1. And so we know that this sequence is not going to converge, it does not come close to any number in rather any vector in
• 11:30 - 12:00 R2. So, this sequence will not converge. So, if I play this one, you will see it is still oscillating between minus 1 comma 1 and 1 comma 1. And we knew that already, as I explained, because minus 1 to the power n oscillates. So, in other words, this does converge and this converges to 0, 1 we saw this graphically also, this does not converge or this limit does not exist. And now we have
• 12:00 - 12:30 the last one. So, let us look at these terms. So, the first term is cosine n by n. So, cosine n, as we know, varies between minus 1 and 1 and, but when you divide it by n, so that means in absolute value, it is between 0 and 1 by n. And then by sandwich, we know that this tends to 0. The second one we have done
• 12:30 - 13:00 in our previous video on limits, when we did limits for sequences. So, I encourage you to solve it yourself and if you do not remember, go back to that video and ask, see what happens. It is five times something, I will leave that to you. The third one, we saw in the previous slide goes to e. And what about the fourth one,
• 13:00 - 13:30 so the fourth one is n times cosine 1 by n. And unfortunately for this, we know that cosine 1 by n as n tends to infinity, so 1 by n tends to 0, so cosine 1 by n tends to cosine of 0, which is 1, but n goes shoots off to infinity. So, this is going to go to, this is going to diverge to infinity.
• 13:30 - 14:00 And so, what does this mean? That means that in this sequence as a whole, as a sequence in R4, three of these coordinate sequences do indeed have limits, but the fourth one does not. So, this limit also does not exist, limit does not exist. So, I hope it is clear what we mean by sequences of limits in higher dimensions. So, we have to just check individual limits for each coordinate.
• 14:00 - 14:30 So, let us now talk about the limit of a scalar valued multivariable function at a point. So, this is going to generalise what we saw earlier about one variable calculus, limits of functions of one variable at a point. So, let f be a scalar valued multivariable function defined on a domain D in Rk and a tilde be a point such that there exists a sequence in D which converges to a tilde. So, this is some technical condition that we need in order to define the notion of the
• 14:30 - 15:00 limit. So, if there exists a real number L, remember this is a scalar valued function. So, f takes values in R, which means real numbers. So, if there exists a real number L such that f of an tilde tends to L for all sequences an tilde such that an tilde tends to a tilde. Then we say that the limit of a at a tilde exists and equals L. So, this is again, direct generalisation of what we have seen in one variable calculus.
• 15:00 - 15:30 We denote this by a limit extends to a tilde f of x tilde is equal to L. So, this is the same as saying that as x tilde comes closer and closer to the point at a tilde, then f of x tilde eventually comes closer and closer to the number L. This is what what it means for this limit to exist. And if there is no such number L, we say that the limit of f at a tilde does not exist. Let us do some basic examples. So, this first limit is ai to the power k. What is ai? ai
• 15:30 - 16:00 is the ith coordinate of the vector a tilde. So, you look at that and raise it to the power k and it is fairly obvious that that is what happens. Because as x tilde tends to a tilde you want xi to the power k well, x tilde tends to a tilde means xi is coming close to ai, that means xi to the power k is coming close to ai to the power k. Similarly, for the second one, you will get ai to the power k, because here we have assumed
• 16:00 - 16:30 ai is non-zero. So even if your power is negative, meaning this k is negative, the same argument will hold, how would e to the power xi, well this will go to e to the power ai. How would log of xi, well again, if ai is positive, this is going to be log of ai. And then sin of xi will go to sin of ai. Please check this, whatever I am, I am doing this fast, because I think this is very doable
• 16:30 - 17:00 from what we have done before. But you should check this for yourself, it is very important that you check these and understand what what is being said here based on the definitions in the previous slides, fine. And then finally, if you have tan of xi and ai is between minus pi by 2 and pi by 2, then this is tan of ai. So, this is not, this is nothing very deep, it just follows from one variable calculus. And what I am going to try and say, tell you now is that most of the things that we do for easy limits are from one variable calculus. There are difficult
• 17:00 - 17:30 limits, of course, and that is, that is what we will be studying as we go along. Those need more refined techniques. So, let us look at some rules about limits of scalar valued multivariable functions. These are again in the same spirit as the ones for one variable calculus. So, if you have two functions such that, for both of them, the limits as x tilde tends to a tilde exists and equal F and G respectively, and you have some scalar c, then if you take cf
• 17:30 - 18:00 plus g, then that limit is c times capital F plus capital G. So, you can push the limit inside and take the scalar out, that is what it says. And of course, special cases are, of this are where c is equal to 1. So, in that case, it will say that the sum of the limits is the limit of the sums. Similarly, you can take c is minus 1 and that will say that if you take the difference of the functions and take the limit, that is just taking the difference of the limits, you can just take the function
• 18:00 - 18:30 g is 0, and in that case, this is saying c times f limit, you can take the c out. So, the next one is for product. So, if you take the multiplication of two scalar valued functions. And this makes sense because these are scalar valued multivariable functions, then if you take the limit, and both of these limits exist as x tilde tends to a tilde, then the limit is just the product. So, it is F times G. Similarly, if you take the quotient
• 18:30 - 19:00 f by g, here, of course you need the caveat that as x tilde comes close to a tilde this the limit is nonzero. So, this capital G is non-zero. So once that happens, the quotient also has limit and it is exactly capital F by G. So, let us quickly write down what this means based on what we have seen. So, for example, if I have a function like like this, so if I have h of x, y, z is x square y cube plus
• 19:00 - 19:30 y cube z squared plus e to the x. Let us say e to the xyz. And I want to ask what is the limit of let us say x, y, z tends to 1, 2, 3 of h. Well, let us let us see if I can apply
• 19:30 - 20:00 whatever I have above, let me not take e to the power of x, y, z instead let me take just x y, z. We will come to x, y, e to the xyz in a minute. So, if you take this limit, well, first let us look at x square y cube. We saw on the previous slide that both of these individually exists. And this limit is going to be 1 squared the substitution and this limit is going to be 2 cubed, then you can use the second one
• 20:00 - 20:30 and say that for the, for x square y cube, the limit is 1 squared times 2 cube. Similarly, for y cube x squared, it will be 2 cube times 3 squared. And for the third term, it will be 1 times 2 times 3, using 2, and then using 1 that you can, the limit of a sum of functions is the sum of the limits, if all of them exist,
• 20:30 - 21:00 you can just say that this is the limit of each of these individually. And of course, limit x, y, z tends to 1, 2, 3; x, y, z tends to 1, 2, 3 and we have computed all those
• 21:00 - 21:30 using again these rules. So, this is 1 squared times 2 cube plus 2 cubed times 3 squared plus 1 times 2 times 3, essentially, what we have got is you can just substitute h of 1, 2, 3 inside this function. So, you can, this limit is exactly h of 1, 2, 3, which is, so 8 plus 8 times 9 72 plus 6, so whatever that is, that is I think 86.
• 21:30 - 22:00 So, I hope it is, it is clear how these rules are useful. Similarly, you could have something like limit let us say x by y as x comma y tends to 1 comma 1, then that would be just 1, because you can substitute for the numerators, substitute for the denominator, and for the
• 22:00 - 22:30 denominator it works because the limit is non-zero. So, this is very, very, very similar to what we have done in one variable calculus in many, many, many places you are, you can just substitute the values. Not always though, remember even in one variable calculus, we had trouble with substitution, for example, in things like sin x by x. But I hope it is clear how these rules can be used. So, for polynomials, for example, it is very easy to find limits.
• 22:30 - 23:00 The next thing that we want to study is composition. So, suppose f is a scalar valued multivariable function, and g is a function of one variable such that the composition g composed f is well defined. Then, if limit f of x is F, and limit g of x is L, then limit g composed f of x is also L. This may take a little time to digest, what we are saying is this.
• 23:00 - 23:30 So, suppose I have the function h of x, y, z is e to the power xyz. Then I can write this function as a composition of two functions. One is h of x, y, z is xyz. This is a very nice function that we understand, not h, but f of x, y, z and the other function is that is a g of u is equal to e to the power u, again, a very nice function, both of these
• 23:30 - 24:00 are nice functions. Let us see what happens to limit, let us say I want to compute limit x, y, z. So, want limit x, y, z tends to 1, 2, 3 now h of x, y, z this is what I want. Now, whatever we did on the previous page does not unfortunately help us directly. But what we can do is we
• 24:00 - 24:30 can use the fact that this is com, this is a composition. So, let us first ask what happens to f? Well, this, of course, we know what happens this is 1 times 2 times 3. So, in terms of what is your, this is capital F and, and then if we have limit x tends to f of g of x, so here, we want u tends to this number which is 6, so u tends to 6 g of u,
• 24:30 - 25:00 which is e to the power u, this is e to the power 6. So, what is the net result? The net result is that this limit that we wanted is exactly this number here, e to the power 6
• 25:00 - 25:30 because I can compose. So, that is the answer. And this is again, eventually what we are saying is you can substitute. So, this is a useful rule, then we have functions beyond just polynomials. For polynomials the previous page, whatever we had or rational functions, the previous page sufficed. But if you have exponentials, trigonometric functions, logarithms then we can use this composition rule and we can find the limits. And finally, we have the sandwich principle, which again, we have studied in one variable
• 25:30 - 26:00 calculus. So, if you have two functions f and g such that for both of them, the limit as x tends to a x tilde tends to a tilde is L. And then if you have some function which is caught between them, sandwiched between them, then the limit as x tends to a tilde h of x tilde is also equal to L. So, you will see an example of this in the tutorial. So, I will not expound more on this. But we have seen examples already in one variable calculus.
• 26:00 - 26:30 So let us come to finding limits by substitution. And this is where you have to be fairly careful. So, suppose we want to find the value of the limit of a function f of x at the point a tilde so that is limit x tends to, x tilde tends to a tilde f of x tilde. So, often we can substitute the value of a tilde in the expression of f of x tilde and obtain the limit. This is what many of the examples that we have seen, we could do.
• 26:30 - 27:00 So, unfortunately, this does not work, when the function gets slightly complicated, or the point a tilde does not belong to the domain of the definition of f of x tilde. It may happen that f is not defined a tilde at all, then what do you do? So, let us look at this example, you may actually remember this example from somewhere else. In any case, let us try and understand what is limit x tilde tends to 0, 0 x cubed minus y squared x by x squared plus y squared, squared. So, what we want to ask is, what happens as,
• 27:00 - 27:30 to this limit as x tilde comes close to 0, 0, which means x, y comes close to 0, 0. Now, we cannot use any of the previous things because if you look at this expression, it is a 0 by 0 type expression, so you cannot, so the rational function where you had F by G, unfortunately,
• 27:30 - 28:00 the denominator here is becoming 0 in the limit, so you cannot use that rule. So, there is no other rule that we can really use. So, how do I try to attempt this? So, we have to sort of try and do this by first principles, ahead we will also see other ways. So, we will try and look at what happens to sequences as they come close to 0, 0. So let us first try and look at sequences of the form, let us say I look at a sequence of the form an tilde is equal to 1 by n, 0.
• 28:00 - 28:30 So, along the x axis as I come close to this point 0, 0, what happens to this function? So, if you look at f of an tilde, well, I can substitute 1 by n, 0 in this expression, and what do I get? So, the numerator is 1 by n cubed minus 0 squared times 1 by n. And
• 28:30 - 29:00 the denominator is 1 by n squared plus 0 squared, the whole squared. So, this is, if we compute this, this is 1 by n cubed divided by 1 by n to the power 4 which is just n. And what that means is, if you take a f of an tilde, that is exactly
• 29:00 - 29:30 n, so f of an tilde as n increases, diverges to infinity. So, this function in terms of what we have seen, there is a sequence for which this function diverges to infinity. Just for so, so in particular, what this means is that the sequence does not have a limit. So, this limit x tilde tends to 0, 0 does not exist of this function. Let us just for
• 29:30 - 30:00 practice, take another sequence was so 0, 1 by n. So let us look at what is f of bn tilde, so for bn tilde if we substitute that, so the numerator is 0 cubed minus 1 by n squared times 0, that is 0 and the denominator is 0 squared plus 1 by n squared squared. So, which is 0 by something non-zero which is just 0.
• 30:00 - 30:30 So, for this, this sequence bn tilde f of bn tilde does actually have a limit. So, along the y axis, this function does have a limit. So, notice what is happening here along the x axis as we take a sequence along the x axis which was 1 by n comma 0 the f of an tilde diverged to infinity along the y axis as we took a sequence that actually converges to
• 30:30 - 31:00 0. So, we have two sequences which are giving us two different limits. So, there is no way in fact, one of them diverges. So, this limit does not exist. So, the main point is you cannot substitute, this for example, in this kind of function, you cannot substitute and this limit actually does not exist. So, this idea of taking sequences from different directions
• 31:00 - 31:30 is very important and we will see more about this ahead. Thank you.