AI Assistant
Transcript
00:00:790Paolo Guiotto: Yeah.
00:01:820Paolo Guiotto: Okay, great.
00:19:900Paolo Guiotto: Okay, so… Let's restart from the point we finished.
00:29:460Paolo Guiotto: Last time… We introduced the definition of limit for a function.
00:37:90Paolo Guiotto: function of vector variable, so the variable is an array, X, that belongs to RD, more precisely to some domain.
00:47:140Paolo Guiotto: of RD, and the function is vector-valued.
00:51:210Paolo Guiotto: So F of X is not a number in general. It could be, but it can be also a vector in RM.
01:00:190Paolo Guiotto: Well, the definition says that the limit of f of x when x goes to P, P, of course, will be a vector of the same environment of X, so a vector of…
01:12:690Paolo Guiotto: RD, and it must be an accumulation point for the domain. We'll return on this in a moment.
01:20:750Paolo Guiotto: It's just the analogous definition you have for accumulation points in the real life.
01:26:00Paolo Guiotto: So we say that the function f goes to limit L, which can be here any vector of the codomain, so of RM, or possibly the infinite of that space.
01:39:430Paolo Guiotto: If the following fact holds, whenever you take a sequence of points X and that belongs to D, different from point P, where you compute limit.
01:50:80Paolo Guiotto: This is required because normally the function f is not defined at the point where you want to compute limit.
01:58:30Paolo Guiotto: Or, it may happen that the function is defined at this point, but the value of the function at that point has nothing to do with the limit.
02:08:509Paolo Guiotto: When the limit coincides with the value, we will have the definition of a continuous function. We will return on this.
02:14:740Paolo Guiotto: Later.
02:16:500Paolo Guiotto: and Xn goes to P, what happens? FXN goes to L. So, no matter how you go to P, in the domain, not touching P, the function f will drive you to L.
02:28:380Paolo Guiotto: That's the definition of limit.
02:30:880Paolo Guiotto: Now, before, We… We precise what does it mean, accumulation point.
02:39:130Paolo Guiotto: Let's just do a remark.
02:42:520Paolo Guiotto: I forgot to copy this thing.
02:46:220Paolo Guiotto: Okay.
02:48:900Paolo Guiotto: So, let's do a remark here on the definition, important remark.
02:59:400Paolo Guiotto: So, we, we say, That, let me just, repeat. There exists the limit.
03:10:890Paolo Guiotto: for X going to P.
03:14:620Paolo Guiotto: of… F.
03:16:720Paolo Guiotto: of X.
03:18:580Paolo Guiotto: equal L.
03:20:940Paolo Guiotto: If and only if… for every sequence, XN.
03:26:350Paolo Guiotto: contained in domain D.
03:28:820Paolo Guiotto: Xn different from limit point P.
03:33:430Paolo Guiotto: and XN.
03:35:800Paolo Guiotto: going to P.
03:38:340Paolo Guiotto: Then The sequence of vectors F at point XN, goes to vector n.
03:48:630Paolo Guiotto: In particular, from this definition.
03:51:370Paolo Guiotto: it follows a test to check that the limit actually does not exist. That would be quite important. So, in particular.
04:05:890Paolo Guiotto: We… However… that.
04:10:730Paolo Guiotto: So, if this is… if this property here
04:14:790Paolo Guiotto: If this property is false, is not verified, the limit does not exist. Now, when is this property not verified? This property says, whenever you go to P, the function goes to L. Let's say roughly in this way.
04:31:770Paolo Guiotto: So, this property is not verified when you can find two, at least two ways to go to P, and the function does not go to the same L.
04:42:700Paolo Guiotto: Okay, behaves differently.
04:45:300Paolo Guiotto: So, in particular, we have that if
04:49:630Paolo Guiotto: there existed two sequences, let's call them XN, And, say, YN.
04:58:280Paolo Guiotto: contained in domain, both different from the limit point P.
05:08:640Paolo Guiotto: Both going to… B.
05:19:480Paolo Guiotto: Such that…
05:22:950Paolo Guiotto: The first one, when we evaluate F at point Xn, let's say we go to a limit L1.
05:32:400Paolo Guiotto: And the second one, when we evaluate F at point YN,
05:36:850Paolo Guiotto: we go to another value, L2 vector, different from L1. So with… L1 is different.
05:50:750Paolo Guiotto: from… L2, then in this case, the conclusion is that there cannot be the limiter…
05:59:660Paolo Guiotto: when X goes to P, of… F of X.
06:07:150Paolo Guiotto: Let's say that. The definition is like to say, at least I don't know if this is something which is used in other cultures, but in Italy, we say all the ways go to Rome, huh? Okay, because this was how the approaches were done in the Anson-Roman
06:23:280Paolo Guiotto: kingdom.
06:25:220Paolo Guiotto: So, empire. That's what the definition says. So, if you have two ways that go to Rome, that the function does not go into the same place, this means that the function has not a limit. So it's just a logical
06:40:670Paolo Guiotto: derivation from this property. But keep in mind, because this will be very important to discuss cases when we suspect that the limit of a function
06:53:900Paolo Guiotto: Does not exist.
06:56:190Paolo Guiotto: Okay, now, what we have to do is to start, look… to start to having a look to problems.
07:08:580Paolo Guiotto: Actually, here… There is something, but I will skip this part.
07:14:320Paolo Guiotto: But before we do that, let me just clarify what does it mean that,
07:18:870Paolo Guiotto: P is an accumulation point, which is the requirement, the basic requirement we need to compute any limit. This is not a novelty. Also, in the definition of limit for a function, the point where you compute the limit must be an accumulation point.
07:36:270Paolo Guiotto: Now, this is a technical requirement that must be fulfilled by the point where we want to compute the limit. What does it mean? Well, intuitively, the definition is the following.
07:48:250Paolo Guiotto: Since you want to compute the limit of f of x when x goes to p, you need to be able to compute f of x when x is close to p.
07:56:630Paolo Guiotto: So it means that point P cannot be any point, because it must be somehow surrounded by points of the domain, otherwise of the function, otherwise I cannot compute F of X.
08:08:880Paolo Guiotto: So the condition is something like, habitually close to the point P, there are points of the domain D of the function F.
08:19:380Paolo Guiotto: Now, this can be written in different ways. I prefer here to use sequences, because we have this… I've already introduced this tool. So, let's,
08:32:750Paolo Guiotto: do this, to complete.
08:40:640Paolo Guiotto: the… definition.
08:45:140Paolo Guiotto: of limit.
08:48:00Paolo Guiotto: We need Dude.
08:52:790Paolo Guiotto: Say… What?
08:57:420Paolo Guiotto: does.
08:59:430Paolo Guiotto: It's mean…
09:04:440Paolo Guiotto: that the point P, Is an accumulation point.
09:09:690Paolo Guiotto: for D.
09:11:330Paolo Guiotto: So, the definition is the following definition.
09:16:490Paolo Guiotto: The nice of sequences is that we can use a unique definition for all possible cases.
09:22:440Paolo Guiotto: In the sense that, we say that
09:30:100Paolo Guiotto: Appoint P.
09:31:880Paolo Guiotto: That can be, in this definition, either a point of RM,
09:39:350Paolo Guiotto: I do not, let me in a second go back to the…
09:43:130Paolo Guiotto: Notations… no, we use the RD for the domain, it's just…
09:50:330Paolo Guiotto: So, let's say RD is not important, because RD or RM is the same thing.
09:54:880Paolo Guiotto: We say that the pointer over D, or possibly they infine it alphaD,
10:01:400Paolo Guiotto: So this definition works also for the infinite term.
10:07:340Paolo Guiotto: And… accumulation.
10:15:380Paolo Guiotto: point.
10:18:450Paolo Guiotto: 4.
10:20:330Paolo Guiotto: D, contained into RD.
10:25:300Paolo Guiotto: If…
10:27:290Paolo Guiotto: Well, the condition is that we want to say that there are lots of points of D close to P. We can say this in the following wave, if there exists a sequence
10:40:10Paolo Guiotto: XN, at least one.
10:42:830Paolo Guiotto: sequence, of points of D, such that this sequence XN is different from point P. So, this is needed because, this necessarily will mean if we,
10:59:810Paolo Guiotto: considered together with this second condition, that XN must go to P.
11:04:300Paolo Guiotto: So if you have to imagine a sequence of this type, let's do a figure like that. Let's say that this is the domain D,
11:11:680Paolo Guiotto: For example, the domain D is the black region here. This point here, P,
11:18:770Paolo Guiotto: is not an accumulation point of D, because you have to imagine a sequence of black points… let's use maybe a different color, red, here. We may image… we have to imagine if there exists a sequence of black points here that go
11:35:420Paolo Guiotto: that goes to the point P, which is impossible, because if you are in the black region, and you want to reach the red point, sooner or later, you should get out of there, no? So this is not an accumulation point. But if I take a point like here, this point P, this is now an accumulation point.
11:55:240Paolo Guiotto: of D, because, you see, there is at least one sequence, there are infinitely many, actually, of points of D that go to D, not touching the point D itself.
12:06:840Paolo Guiotto: For example, D might be of this type. Imagine that this is D, is this island, plus a point like this one. Suppose that this point, P, is in the domain. So the domain is made of the island, plus that single point, isolated point, somewhere.
12:25:970Paolo Guiotto: Now, is this point an accumulation point for D?
12:30:440Paolo Guiotto: Now, if you look at the definition, and that explains why there is this condition.
12:36:340Paolo Guiotto: If we have to think, is this point in the… if we have to answer to this question, is this point surrounded by points of D?
12:44:650Paolo Guiotto: We may say no, because he is Indy, so that point is Indy. But there is nothing else around him, no?
12:53:910Paolo Guiotto: So we would say that this is an isolated point, and in fact, if I have to take… if I have to find the sequence of points of D different from point P,
13:04:340Paolo Guiotto: that go to P. The only possibility is that I imagine a sequence in this island that goes to this point, which is impossible, because to go to this point, I should get close to this point, and I should be somewhere here.
13:17:440Paolo Guiotto: In that case, I wouldn't be in D, okay? If I remove that constraint, and I just write the definition with, let's imagine for a second that we just cancel this, and we take this as definition of accumulation point.
13:35:360Paolo Guiotto: I say that point P is an accumulation point for D, if there exists a sequence of points of D such that XN goes to P.
13:42:350Paolo Guiotto: With that condition, this would be an accumulation point.
13:46:260Paolo Guiotto: Because in that case, I would take a sequence Xn constantly equal to P.
13:50:760Paolo Guiotto: No? That sequence, PPPP, forever, is going to be, yes, it is a constant sequence, it stays all time there, so it goes to P.
13:59:960Paolo Guiotto: And it fulfills that condition. But I wouldn't think that this is an accumulation point, because the idea of accumulation point is that there are lots of points of the set D around the point P, not just the point P alone.
14:15:430Paolo Guiotto: That's why we need to have that condition. That's not something that depends on the fact that we are working with vectors, because it's the same for numbers, okay?
14:27:810Paolo Guiotto: even for… if you eliminate the arrows, you have this… the definition of accumulation point for a domain of R. Okay, so an accumulation point is just a point which is, let's say, intuitively surrounded by
14:42:620Paolo Guiotto: points of the domain different from point P.
14:45:790Paolo Guiotto: Okay?
14:47:400Paolo Guiotto: This is also… so this is to say that this point, P,
14:51:930Paolo Guiotto: is not an accumulation point for D. But, for example, let's show another example. Imagine that
15:00:190Paolo Guiotto: In these examples, I have drawn the set
15:06:40Paolo Guiotto: D… with that continuous black line.
15:10:190Paolo Guiotto: that represent the boundary. We have not yet introduced this concept that, let's say, the boundary of the domain D. We may say that, what if we eliminate the boundary? So we have a domain D without…
15:24:820Paolo Guiotto: The skin, let's say, it is the region entirely contained in data.
15:30:360Paolo Guiotto: Thing. With this dashed line, I mean that these points are not in there. Now, if you take this point, a point on that line.
15:40:670Paolo Guiotto: Here, this pointer P is not in D.
15:45:390Paolo Guiotto: But it is an accumulation point, because you can imagine that there are sequences of points of D that goes… that could go to him. So this is an accumulation point for D.
15:58:730Paolo Guiotto: So this is to say that accumulation point does not necessarily mean that we are in the domain D, okay?
16:06:570Paolo Guiotto: And in most of the cases, for calculus of limits, the point where we compute the limit must be an accumulation point, and in most of the cases, won't be in the domain of the function.
16:17:440Paolo Guiotto: This is because one of the reasons we compute limits is because we want to know what happens to a function when we… when the variable goes somewhere where the limit… the function itself does not make any sense. So we want to see what happens.
16:32:930Paolo Guiotto: We will not spend so much time on this, we will check. Well, there is some exercise at the end of the chapter, maybe I will…
16:42:850Paolo Guiotto: I'll let you know some, example.
16:46:70Paolo Guiotto: Okay, let's return on the definition of limit. So, we have this definition.
16:51:760Paolo Guiotto: We have this remark, important remark, that is useful to disprove the existence of a limit.
16:59:940Paolo Guiotto: And, of course, as the definition of limit, you have already seen this in the first year calculus.
17:08:859Paolo Guiotto: We do not compute limits by using the definition, okay? But we have to start from some point, so what I would like to do now is to try to see what happens on some concrete case, no? Then we try to understand some…
17:25:790Paolo Guiotto: general mechanism. So, to begin, so we take the… an empirical road. Let's see what happens when we try to apply these definitions and a few facts we have seen. So, let's start with this example, which is the example 141.
17:45:430Paolo Guiotto: Here's a one-star example, so one star means not particularly hard technicalities, but it is interesting to
17:55:340Paolo Guiotto: to… so you don't have to focus too much on calculations. So, this class… Say… existence…
18:09:550Paolo Guiotto: And… value.
18:15:890Paolo Guiotto: Anyk?
18:18:120Paolo Guiotto: of the limit. So most of the examples will be for functions of two, three variables. Do not make the situation too much complicated.
18:30:510Paolo Guiotto: and real-valued, okay? So we start focusing on this. So here we have a limit. Instead of writing arrays, I will write the vector with the components. So the vector is XY that goes to 0, 0 here.
18:48:380Paolo Guiotto: of this quantity, X times Y divided X squared plus Y squared.
18:57:280Paolo Guiotto: Okay, let's start working on this problem.
19:00:800Paolo Guiotto: Let's first, since this is the first time we have something like this, let's start from, really from, zero.
19:09:750Paolo Guiotto: So, first of all, this is the function we have to compute the limit, so let's discuss a bit this. So, let FXY is our function.
19:21:540Paolo Guiotto: As you can see, I'm not using arrows.
19:24:860Paolo Guiotto: the variable of DSF is a vector, XY, but the value is a number, okay? You see? This is a numerical function, that's why there is not the array.
19:37:870Paolo Guiotto: the arrow, above F. So this is defined in this way, X times Y divided X squared plus Y squared.
19:46:400Paolo Guiotto: I can imagine that this is defined on the natural domain. The natural domain is the set of XY for which this thing makes sense, okay?
19:55:230Paolo Guiotto: defined for XY in D, where…
20:01:590Paolo Guiotto: Now, what is D? D is the set of points, X, Y, or vectors, if you prefer, in R2, such that
20:11:280Paolo Guiotto: this quantity makes sense. Now, X times Y clearly makes sense, whatever X and Y are. X squared plus Y squared makes sense, whatever this X and Y.
20:23:140Paolo Guiotto: Then there is a fraction, and the unit thing I need to check if the denominator is different from 0. So the unit problem is when denominator is equal to 0. So, the point XY net2, such that X squared plus Y squared is different from 0.
20:42:610Paolo Guiotto: Now, let's see what is X squared plus Y squared equal to 0. Well, that's easy because we have the sum of positive quantities, which is 0. This is possible if and only if both are equal to 0.
20:58:730Paolo Guiotto: So, this means that the domain contains all possible XY except one single point, which is the point . So, this means that D is made of, all XY
21:12:770Paolo Guiotto: In R2, Such that the point XY is different from point 00.
21:22:480Paolo Guiotto: Well, be careful, because this is wrong. X equals 0 or Y equals 0. Okay, what is not allowed is that both
21:32:820Paolo Guiotto: are equal to zero. So, if I, say…
21:38:830Paolo Guiotto: if I write this X different from… X different from 0, and Y different from 0, or this, X different from 0, and Y different from 0. Is that correct?
21:54:370Paolo Guiotto: So, this one…
21:56:440Paolo Guiotto: means one of the two is different from zero. This means both of two are different from zero.
22:05:430Paolo Guiotto: Are they the same of this? Is one of them the same of this?
22:12:110Paolo Guiotto: Now, be careful, because the first one says.
22:15:560Paolo Guiotto: X different from 0, or Y different from 0?
22:21:40Paolo Guiotto: So, the second one.
22:24:280Paolo Guiotto: You think that the second one is true?
22:29:890Paolo Guiotto: No, but I'm giving condition on the components of the vector. Let's say that what is written is correct. I'm asking you.
22:39:790Paolo Guiotto: Is that the same of saying XY different from 00?
22:51:190Paolo Guiotto: If one of them is zero… not zero, that's, you know.
22:54:520Paolo Guiotto: Yeah, so which one is correct? Both are correct. Okay, one answer is both are correct.
23:01:660Paolo Guiotto: Any other opinion?
23:04:170Paolo Guiotto: You say the second one is correct or wrong.
23:08:170Paolo Guiotto: So, someone says that this is correct, someone says that this is correct.
23:14:350Paolo Guiotto: Any other opinion? Both are wrong. Both wrong. Okay.
23:19:910Paolo Guiotto: Sure, and that… Number one, correct.
23:26:480Paolo Guiotto: Number two, correct?
23:30:600Paolo Guiotto: Then, both… Wronged…
23:39:600Paolo Guiotto: Any other opinion?
23:43:40Paolo Guiotto: Okay, let's do all the possible combinations, no? Both correct.
23:52:880Paolo Guiotto: Okay, let's think about… So, it says that we are in the domain D if X squared plus Y squared is different from 0. So, we are not in domain D, if X squared plus Y squared is 0, that means both X and Y are equal to 0.
24:09:930Paolo Guiotto: So both X and Y equal 0 means you are not in D.
24:13:480Paolo Guiotto: So definitely, now, what does it mean with respect to these two properties?
24:20:240Paolo Guiotto: So, the first one says, this is the number 2.
24:24:860Paolo Guiotto: X different from 0, or Y different from 0? So, at least one of these two things must be true.
24:33:510Paolo Guiotto: And this means that they cannot be both equal 0. So this one is, in fact, the same. The number 1 is true.
24:44:990Paolo Guiotto: About number 2, X different from 0 and Y different from 0 means both different from 0.
24:52:240Paolo Guiotto: So, for example, a point like 0, 1,
24:57:20Paolo Guiotto: do not verify… does not verify this second condition. So, apparently, it is not in the domain, but it is in the domain, because if you put… you see that it is not true that both components are zero. So, that's wrong.
25:15:930Paolo Guiotto: And of course, all these are wrong. So the characterization could be this one, the first one.
25:24:670Paolo Guiotto: Now, you can see here, we can even do a little figure, because we are in the Cartesian plane, XY, and what we are saying is that we take all points.
25:38:180Paolo Guiotto: Except the point with X and Y equals 0. What is this point? It is the origin.
25:44:660Paolo Guiotto: So the domain is everything except that red point.
25:50:300Paolo Guiotto: Okay, so I can say that D is the full plane, R2 minus… do you know this symbol? Okay, the set made of .00.
26:03:830Paolo Guiotto: So you take out that single point.
26:07:790Paolo Guiotto: Okay, so this is the, the definition of F, the domain, yeah?
26:14:320Paolo Guiotto: Yeah, but if we say that the second is wrong, they cannot be both correct.
26:23:230Paolo Guiotto: You see?
26:26:490Paolo Guiotto: That's… from a logical point of view, I don't know what is… okay.
26:31:200Paolo Guiotto: So, now… Let's see what is written here about the limiter. So… We have,
26:40:860Paolo Guiotto: We, we have… to discuss… the limit…
26:50:420Paolo Guiotto: for XY going to 0, 0. So this 00 is just our point P in the notations of the definition of the function F.
27:01:930Paolo Guiotto: So, in principle, the first thing we should check
27:07:550Paolo Guiotto: is, well, let's go back to the definition, the complete definition. We should check this.
27:14:880Paolo Guiotto: Is P an accumulation point for the domain? Because if it is not an accumulation point, the operation limit does not make sense, okay? So why we'll do just once? Because we assume that whenever we write a limit, it makes sense. But be careful, because in principle, you should always…
27:34:980Paolo Guiotto: Check this.
27:36:390Paolo Guiotto: So… Let's check.
27:43:780Paolo Guiotto: that,
27:46:670Paolo Guiotto: P is an accumulation point for the domain, which is, in this case, let's say, evident, because look, the domain is the full plane except the red point. Now, so the domain is all this.
28:02:560Paolo Guiotto: it is clear that the point, right, is surrounded by black points, no? For example, if you want a concrete example, just once.
28:12:120Paolo Guiotto: and forever, is we could take a sequence like this of points moving on the x-axis, they go to the point .
28:22:900Paolo Guiotto: Now, let's write these sequences. Since they are on the x-axis, they will be something like 0.
28:29:300Paolo Guiotto: They are of this type. Now, I need to put an abshisha that makes the point moving to the left to 0. So, for example, I could put 1 over N,
28:42:130Paolo Guiotto: So if I have these points, these are… So, 1,0… one half Zero.
28:52:160Paolo Guiotto: one-third.
28:54:520Paolo Guiotto: 0, and so on. Of course, there are infinitely many, I cannot plot all of them, but let's say that, in general, this is the point 1 over n0.
29:05:250Paolo Guiotto: Now, if you are not convinced that this is correct, so let's formally do the check. Let's call XN vector the vector 1 over N0.
29:17:530Paolo Guiotto: Now, is it in the domain D? Yes, because at least one of the two components is different from 0, so it is in domain D.
29:27:770Paolo Guiotto: Clearly, Xn is different from the limit point P, which is, in this case, the vector 0. Reminded, this is our P here.
29:36:950Paolo Guiotto: Because to be equal to zero, you must have all coordinates equal 0, and the first one is definitely different from zero.
29:44:460Paolo Guiotto: And number 3, Xn, when n goes to infinity, goes to 0, 0. Yes, because you can do component by component, the limit, no? 1 over n goes to 0, 0 is constantly equal to 0. It goes to 0, so you see that the two components, they both go to zero.
30:03:700Paolo Guiotto: And therefore, the vector goes to the vector 0, 0. Okay? So you see the three conditions.
30:11:420Paolo Guiotto: That,
30:14:660Paolo Guiotto: are verified. We have a sequence of points of D, different from the limit point P, that go to the limit point P.
30:25:20Paolo Guiotto: And that's exactly what we have in here.
30:28:130Paolo Guiotto: Okay, so this says that 00 is an accumulation point for our domain, and… Limit.
30:40:910Paolo Guiotto: operation… mates… sense. So this does… this…
30:50:00Paolo Guiotto: does not mean that the limit exists, it is well-defined. It means that we can… it makes sense to compute the limit, okay? Otherwise, it would… it wouldn't be an operation without any sense if the limit point is not in the accumulation.
31:06:450Paolo Guiotto: points of the domain. Okay, now we have to enter, really, into the problem of computing this limit.
31:13:440Paolo Guiotto: Now, we always have these alternatives.
31:16:780Paolo Guiotto: Either we prove that the limit exists and it is equal to something.
31:21:700Paolo Guiotto: Or, we prove that the limit does not exist.
31:25:400Paolo Guiotto: Okay?
31:26:700Paolo Guiotto: So, in general, when a problem is posed like that, you don't know, of course, which one… which is the case. So, you have to do something to understand what is more reasonable. Is it reasonable that the limit exists? I try to prove that the limit exists, and this will take…
31:45:230Paolo Guiotto: It's a number of strategies.
31:47:660Paolo Guiotto: Or, do I believe that this limit won't exist? I will try to prove that the limit does not exist by…
31:55:830Paolo Guiotto: using a strategy like that, finding two ways to go to zero on which the function has two different limits. Okay?
32:07:340Paolo Guiotto: So, let's see now what can we say. We are doing some sort of experimental way. So, we have at the limit…
32:17:730Paolo Guiotto: for XY going to 0, 0.
32:21:300Paolo Guiotto: of this X times Y divided X squared plus Y squared.
32:26:310Paolo Guiotto: Now, when you compute a limit, even in one variable, the first thing to do is to understand if there is some trouble, no? Troubles are called indeterminate forms in the whole limit.
32:36:720Paolo Guiotto: Now, XY is going to 0, so what can I say about X and Y?
32:43:320Audio shared by Paolo Guiotto: Well, the point XY goes to the point 00.
32:49:190Paolo Guiotto: I can notice that point XY goes to .00,
32:54:330Paolo Guiotto: If and only if both X and Y are going to 0.
32:59:580Paolo Guiotto: No? Component by component.
33:02:140Paolo Guiotto: So if they are going to zero, I can reasonably understand, I hope that you agree with me, that their product will go to zero.
33:11:350Paolo Guiotto: Okay? So this is going to zero, and X squared is going to 0, Y squared is going to 0, X squared plus Y squared will go to 0. So that's exactly the problem, because I have what is called an indeterminate form.
33:27:30Paolo Guiotto: So, something that, does not follow any rule. So, we have to solve, directly by eliminating this indeterminacy, somehow.
33:40:200Paolo Guiotto: Okay, so that's for sure that we have a problem here. We cannot apply any expected rule.
33:48:80Paolo Guiotto: What else can be observed here?
33:52:10Paolo Guiotto: Well, when you have a form like 0 over 0 for one variable limit.
33:57:290Paolo Guiotto: To solve the indeterminacy, what you do normally is to try to understand how much zero is the numerator, how much 0 is the denominator.
34:06:430Paolo Guiotto: So, there is an idea behind it, which is the idea of order of zero. So, if this is more zero than this, it's like if this number is smaller than this one, the fraction is going to be small, so the limit is going to be zero.
34:22:60Paolo Guiotto: Or, if this one is smaller than this, it means that it's like if you are dividing by 0, so the fraction will be big. So that's Y is an indeterminate form.
34:31:770Paolo Guiotto: Now, if you look at these two, it's difficult here, because there are two variables.
34:37:409Paolo Guiotto: Now, if you have X square, only this one, and X, you have a first degree zero, a second degree zero. X is going to zero. In fact, X divided X squared would be 1 over X, and it goes to infinity, something like this.
34:53:790Paolo Guiotto: Okay? But since we have X and Y, X times Y, and both are going to 0, it is not clear how much is 0 the numerator.
35:03:590Paolo Guiotto: If you look, they are polynomials, but polynomials in two variables.
35:09:280Paolo Guiotto: And they are both, like, second degree stuff, because this is, there are squares. Here, there is a product of first degree, so it's something like a second degree.
35:18:560Paolo Guiotto: So, if I should bet on this, I would say they should have more or less the same magnitude. So, I may expect that
35:28:90Paolo Guiotto: Perhaps the limit is different from zero.
35:31:640Paolo Guiotto: But look what happens.
35:33:670Paolo Guiotto: if I, restrict my focus on spatial points.
35:39:490Paolo Guiotto: So, this figure is a figure on the domain, so we are not going to do any figure for the values of F.
35:47:690Paolo Guiotto: This is complicated, because to plot even this function, we need to do a shape in R3. Two axes for the domain, and one axis for the numbers, the F of XY.
36:02:390Paolo Guiotto: So typically, a plot is made like that. You can do by a computer. You have two axes for X and Y, and the axis is used for the values FXY. So for every point here, you will have a number, and then you have a point, XY, and the third coordinate will be FXY.
36:22:620Paolo Guiotto: So the plot of this thing will be something like a surface.
36:27:160Paolo Guiotto: in R3. It's a complex object.
36:30:900Paolo Guiotto: We do not have tools to do this elementary. In an elementary way.
36:36:220Paolo Guiotto: We could… it's not mine. Okay. We could ask to a computer to do this kind of plot, but we understand that as soon you have a function, for example, of three variables, X, Y, Z, the plot belongs to R4. We cannot ask anymore to the computer to do that plot, so…
36:55:950Paolo Guiotto: forget the way of doing figures. Here we do a figures for the domain.
37:01:850Paolo Guiotto: Now, imagine that you are moving on the x-axis, so take a point that here has shape X0.
37:08:820Paolo Guiotto: And you send this point to zero. So, along the axis, in this way.
37:14:660Paolo Guiotto: Okay, so let me color, maybe… This.
37:18:500Paolo Guiotto: So when pointer X0 goes to 0.00?
37:23:630Paolo Guiotto: Well, this if, no if…
37:30:990Paolo Guiotto: X goes to zero, no?
37:33:580Paolo Guiotto: Okay, so pointer as it is naturally, huh? X0 goes… moves to 0, if and only if X, you send up Shissa to 0. Okay.
37:44:300Paolo Guiotto: Now, look what happens when we evaluate F at these points. This is X times Y, which is 0, divided by…
37:54:680Paolo Guiotto: X squared plus 0 square.
37:58:510Paolo Guiotto: Okay? Notice that since here we are evaluating F on a point of the domain, we are also assuming that this X is different from 0. Otherwise, we would evaluate F at 0, but F is not defined at 0, 0. So this is not a load.
38:14:820Paolo Guiotto: Now, if you compute this, you get X times 0, which is 0, whatever is X, divided X squared.
38:20:750Paolo Guiotto: And since, the, X square at the denominator is different from 0, this is 0.
38:27:160Paolo Guiotto: This is whatever is X.
38:29:450Paolo Guiotto: So when you evaluate F at these points, you get always zero.
38:34:320Paolo Guiotto: So you may think that, since 4… X going to zero.
38:44:580Paolo Guiotto: we have that FX0 is constantly equal to 0. It goes then to zero.
38:52:660Paolo Guiotto: Okay? We may think that along this axis, so evaluating F along these spatial points, I get that my F goes to 0.
39:03:670Paolo Guiotto: Is that sufficient to conclude that the limit when XY goes to 0, 0 of F is 0,
39:10:910Paolo Guiotto: Well, it seems a bit exaggerated, because this is a limit when XY goes to 0, 0 in all possible ways, no matter how XY go to 0. And this is a very special way. However, it suggests that along these points, we go to 0.
39:28:170Paolo Guiotto: A similar argument is for points on the other axis.
39:32:10Paolo Guiotto: So here we have 0.0y, Of course, you expect that 0.0y goes to 0.00 if and only if…
39:40:620Paolo Guiotto: Y goes to zero.
39:43:240Paolo Guiotto: So you have to imagine that now this point is moving. Of course, the point could be also down here. Sorry.
39:52:200Paolo Guiotto: Wow Whatever.
39:58:270Paolo Guiotto: Okay, so the point could be also here, also this one is a 0Y.
40:03:800Paolo Guiotto: In this case, we would… my room.
40:09:380Paolo Guiotto: idiot.
40:12:850Paolo Guiotto: In this case, we will move upward, let's say.
40:16:190Paolo Guiotto: Now, what happens when we evaluate this function f on point 0y? Well, it happens exactly the same thing, because now we have 0 times y divided 0 square plus y squared.
40:29:590Paolo Guiotto: Also, in this case, since 0.0y is not 0, Y is different from 0, so there is no issue with this, so we get 0 over Y squared, it is equal to 0.
40:43:240Paolo Guiotto: So whenever we evaluate the function on the y-axis, we get 0, and so this means that F of 0Y
40:54:80Paolo Guiotto: We go to 0 when Y goes to 0.
40:58:530Paolo Guiotto: So we see the same thing happens even on the y-axis.
41:02:110Paolo Guiotto: So, if we restrict XY along the x-axis, or the y-axis, and we go on this axis to 0.00, the function goes always to 0.
41:14:750Paolo Guiotto: So along these two ways, the two axes, the function goes to zero.
41:20:250Paolo Guiotto: Will that be sufficient to conclude that no matter how we go to 0, 0, the function goes to 0?
41:29:00Paolo Guiotto: Now, it's just two roads, and there are infinitely many roads in plane, okay? And in fact, look at what happens if we now take a third road, which is a straight line, like this one.
41:42:370Paolo Guiotto: Now, this is a line like Y equals MX, right? A straight line passing through the origin. So, a point on this line has an abscessa X and an ordinate, which is MX. So, these are points of type XMX.
42:01:810Paolo Guiotto: Okay? Now, to be different from the origin, it is needed and sufficient that X be different from 0, so X different from 0.
42:13:210Paolo Guiotto: Now, when this point, XMX, M here is a constant, okay? Is any R a real number, fixed?
42:25:330Paolo Guiotto: Okay, is the angular coefficient of the straight line.
42:28:330Paolo Guiotto: Now, this point goes to 00 when…
42:34:110Paolo Guiotto: AXA?
42:35:580Paolo Guiotto: X goes to zero, no? X is the univariable. M is the parameter fixed.
42:40:440Paolo Guiotto: And what if we evaluate F at this point.
42:44:570Paolo Guiotto: At point XMX, the formula says X times Y, so X times MX, divided X squared plus Y squared, MX squared.
42:55:440Paolo Guiotto: Now, this time, the calculation is more interesting, because we get M, X squared numerator. Numerator, you see, there is an X squared, can be factorized to 1 plus M squared.
43:08:80Paolo Guiotto: Now, since our X is here different from 0, we can simplify, otherwise this would be 0 divided 0, so it would not make any sense, so we can simplify this X squared. And what we see is this, we get M divided 1 plus
43:25:320Paolo Guiotto: M squared, which is actually independent of X, it's a constant in X. So it means that when you evaluate F on these points, whatever, wherever they are.
43:37:330Paolo Guiotto: on this straight line, we get always the same value, which is this quantity M divided 1 plus M squared. So, for example, F of XX, I am evaluating along the line Y equal X, is equal to M is 1,
43:56:210Paolo Guiotto: is equal to 1 half, constantly.
44:00:560Paolo Guiotto: And so, yeah, you see something interesting, because, when, X goes to 0, point XX…
44:11:460Paolo Guiotto: Goes to zero. Zero.
44:14:760Paolo Guiotto: But… well, and… the function f… On this point.
44:24:200Paolo Guiotto: Is constant equal to 1 half, it goes to 1 half.
44:29:290Paolo Guiotto: It does not go to zero.
44:32:920Paolo Guiotto: So you see that, huh? If we go to zero along the 0, the origin, along the ax or the y-axis, function f goes to 0. Actually, it is constantly equal to zero.
44:44:790Paolo Guiotto: But if we go to 0 along these lines, y equal mx, we see that the function is constantly equal to something.
44:52:600Paolo Guiotto: So, and even more, FXMX, similarly, is constantly equal to M divided 1 plus M squared.
45:03:720Paolo Guiotto: that goes to itself, M divided 1 plus M squared.
45:08:380Paolo Guiotto: So, as you can see, when you change M, you get a different limit.
45:12:630Paolo Guiotto: So, this is saying, This is the plane XY.
45:18:90Paolo Guiotto: This is our limit point, P.
45:21:10Paolo Guiotto: If we move along this axis, our F here, F is constantly 0, it goes to 0.
45:29:510Paolo Guiotto: But if we move along a line, Y equals MX, and we go to 0 with the pointer here, XY,
45:38:140Paolo Guiotto: F here is constantly equal to that value, M divided 1 plus M squared, and for different M, so for different roads, I have a different limit.
45:52:40Paolo Guiotto: So, this is what? This is exactly this situation. If you want, I will write once and forever. I will show you that we can exactly write everything in this form. So, we have different ways
46:09:810Paolo Guiotto: to go to the point P along which the function has different limits. So this means that the limit won't exist.
46:17:560Paolo Guiotto: Okay, so if you want to see formally.
46:20:650Paolo Guiotto: We will do just once… this is sufficient, the argument we have done, no? Because it is saying, you move along the x-axis, you go to zero. You move along this diagonal straight line, and the function goes to that value M divided 1 plus m squared, different from 0.
46:41:370Paolo Guiotto: formally.
46:44:80Paolo Guiotto: If you want to see the sequences.
46:46:610Paolo Guiotto: I could say, if I take the sequence of points, like, for example, move on the axis 1 over n0,
46:54:590Paolo Guiotto: Here, you evaluate F on 1 over n. 0, you get constantly 0, it goes to 0.
47:02:390Paolo Guiotto: Now, take the sequence YN,
47:05:780Paolo Guiotto: of vectors along, for example, the line Y equals X, so you just need to put the two coordinates the same, so 1 over n, 1 over n. Here, the function on these points
47:20:760Paolo Guiotto: So these are points of type XX, we computed, and we got 1 half.
47:26:550Paolo Guiotto: So here, the function is constant equal to 1 half.
47:30:320Paolo Guiotto: And therefore, it will go to 1 half. So you see there are two different limits along with two different ways to go to the same point. So this means that the conclusion is there is no limit
47:45:990Paolo Guiotto: for XY going to 0, 0.
47:50:140Paolo Guiotto: of the function F.
47:53:470Paolo Guiotto: No, we cannot compute anything here.
47:57:180Paolo Guiotto: Okay.
47:58:780Paolo Guiotto: Do you have any question?
48:01:70Paolo Guiotto: Of course, we are going to repeat many times, so if it is not 100% clear, it's normal. So, let's see in some other example if your comprehension gets improved. Do we want to take a short break?
48:17:750Paolo Guiotto: 5 minutes, okay?
48:53:210Paolo Guiotto: Okay, so this second example is similar.
48:58:440Paolo Guiotto: But a bit more tricky, and it shows us something.
49:04:870Paolo Guiotto: So, well, here, the question is, the following, show that… the limit…
49:16:420Paolo Guiotto: or XY going to 0, 0.
49:21:10Paolo Guiotto: Now, the function looks very similar to the previous one.
49:25:890Paolo Guiotto: XY square, divided by X squared plus Y power 4.
49:33:400Paolo Guiotto: So, show that this limit does not exist.
49:37:840Paolo Guiotto: So, the exercise gives already what is the way we have to do. So, we have to disprove the existence, so we have to find two ways to go to the point 0, 0, along which F has two different limits, okay? So, basically, I repeat, for those who are not listening.
49:57:780Paolo Guiotto: You have to use this.
50:00:750Paolo Guiotto: Okay? Maybe you don't need to use sequences, as we have done here, but you have to find two ways to go to 0, 0 on which F has two different leads. So, let's quickly,
50:14:820Paolo Guiotto: see the situation. So, here the function f is, of course.
50:21:320Paolo Guiotto: this, XY square divided X squared plus Y power 4, divided… defined for XY in a certain subset D of the plane R2, where D is what? Is the set of points, XY,
50:39:380Paolo Guiotto: of R2, such that,
50:43:920Paolo Guiotto: You see, the unit problem is that denominator. The denominator cannot be 0. X squared plus Y power 4, different from 0.
50:53:380Paolo Guiotto: Now, normally is not the best thing to do to discuss when a quantity is different from zero, but it is much easier to discuss when it is equal to zero. So we discuss when x squared plus y power 4 is 0. Here we have,
51:10:80Paolo Guiotto: The same situation as before, this is positive, this is positive, or 0, and the sum is zero, so the unique possibility is that they are both equal 0. So X squared equals Y power 4 equals 0, and this is possible if and if both X and Y are equal to 0.
51:28:270Paolo Guiotto: So, there is a unique point where this thing is 0, it is 0, 0. For all other points, this is never 0. So this means that domain D is everything, the planar 2, except the point 0, which is exactly where we
51:46:760Paolo Guiotto: do have to compute the limit, no? It's not… it's not a casuality.
51:53:670Paolo Guiotto: So,
51:57:610Paolo Guiotto: Of course, we accept this 00 is an accumulation point. You can understand, do a figure. The bad point is where we compute the limit, is this one.
52:08:330Paolo Guiotto: And domain D is everything else. So, clearly, Clearly.
52:14:130Paolo Guiotto: is the same of the previous example. 00 is an accumulation point for the domain D. Okay, so the limit makes sense.
52:24:770Paolo Guiotto: This does not mean that the limit exists.
52:28:60Paolo Guiotto: Now, since the question asks to disprove the existence, we have a… just to look at particular ways
52:37:840Paolo Guiotto: in the plane XY of points going to 0, 0, where F can have different behaviors.
52:47:280Paolo Guiotto: So let's start with the easy ways we have seen before. So we could start with these points, points along the x-axis, or points along the y-axis. Let's see what happens.
53:01:240Paolo Guiotto: So here we have F computing F at point X0,
53:05:700Paolo Guiotto: is X times Y squared, so 0 square, divided X squared plus 0 power 4. Remind that here x will be different from 0, otherwise we would compute the function at 0, which is not allowed.
53:21:730Paolo Guiotto: Now, this is 0 divided X squared, so constantly equal to 0.
53:26:840Paolo Guiotto: So we can say that if we move along the axis, the x-axis.
53:32:370Paolo Guiotto: The point can be at right, at left, okay? Doesn't matter, the function will go to 0 when x goes to 0. What does it mean x goes to 0?
53:46:590Paolo Guiotto: In this probe.
53:50:170Paolo Guiotto: Yeah, but what does it mean? X goes to 0 in this problem? I want a different answer.
53:57:120Paolo Guiotto: Why I wrote when X goes to 0? Because this is equivalent to, say.
54:02:930Paolo Guiotto: yeah, the component, but I don't… I'm not doing a limit in one variable, I'm doing a limit in two variables, so this means…
54:10:530Paolo Guiotto: Point X0 goes where?
54:14:890Paolo Guiotto: This is not… this is why YX goes to 0 is interesting, because this means that that point goes to 0, 0. So if we compute F along this line, well, this line has a name. This is called a section.
54:29:390Paolo Guiotto: Because it is like, the idea, it is, evaluating Evaluating.
54:40:290Paolo Guiotto: F. Along.
54:43:900Paolo Guiotto: So, what have we done here? Well, think about, this is the plane XY. Let's imagine that we can do a figure of the shape of F, so we need the three axes. Two are for the domain, X and Y, and the third one is to represent the values.
55:00:650Paolo Guiotto: When you do the figure of a function y equals f of x, what do you do? You take an x-axis and an orthogonal axis, the y-axis, then for a given X, you plot a point which is XF of X.
55:15:900Paolo Guiotto: A variang X, you get a line, and that line is the diagram of the function.
55:21:340Paolo Guiotto: Now, the same can be done here. The unique difference is that to compute F, I need two values, X and Y. So I need a point down here in the plane XY. I compute the value of F, I get the number.
55:35:680Paolo Guiotto: Now, the triplet XYFXY, so it's a point in space, XYFXY,
55:45:150Paolo Guiotto: is a point in space. Changing XY, I obtain a sort of surface in the plane, okay? So the diagram of a function of this type will be a surface. Normally, we write Z equal FXY, something like this.
56:02:240Paolo Guiotto: Now…
56:03:660Paolo Guiotto: what we are doing here is this. Imagine that we take still the same figure, plane XY, and we evaluate the function f along these points, just these ones.
56:17:380Paolo Guiotto: So forget of all the remaining points.
56:20:200Paolo Guiotto: So, if you look at the graph of F along these points, you will see something like this. By moving the red point, you will see the black point moving along a line.
56:32:180Paolo Guiotto: So this line here is the function F, well, precisely, they would be the point X0, FX0 in space.
56:41:40Paolo Guiotto: But what you see is a line, okay?
56:44:260Paolo Guiotto: It is like if we are slicing the surface here.
56:50:410Paolo Guiotto: And we see a section of the surface. That's why they are called the sections.
56:55:480Paolo Guiotto: So, evaluating F at long, lines.
57:01:300Paolo Guiotto: I don't want to be more formal. There is a definition in notes you can read. We don't really need to be so formal, okay? We complicate things, writing formally everything in this case. So, evaluating F along lines, yields…
57:21:820Paolo Guiotto: What are called the sections?
57:27:170Paolo Guiotto: off.
57:28:770Paolo Guiotto: So a section is a line like that. We don't plot these lines in the examples, we just do the calculation. So, for this example, we evaluate F at point X0, we get that it is constantly equal to 0, so it's a constant line.
57:46:480Paolo Guiotto: Okay, so it is clear that that constant line has a limit, which is the constant itself. So that's the limit for the constant line.
57:54:630Paolo Guiotto: Now, remind our goal. Our goal, since it is clear, we have to disprove the existence, so we need to find a second section where things go differently.
58:08:370Paolo Guiotto: Okay? So we need still to go to the .00 with the argument of the function, but F must go somewhere else.
58:17:500Paolo Guiotto: Now, what happens if we consider, for example, the other axis? So, if we evaluate F on point 0y.
58:25:270Paolo Guiotto: Does it really change? We have 0 times Y squared divided 0 square plus y power 4. That's the evaluation. Remind that y will be different from 0. Why is Y different from 0?
58:45:810Paolo Guiotto: Exactly. We are not computing a function at limit point. We never do that when we compute a limit, okay?
58:53:190Paolo Guiotto: with the remarkable exception, which is the case of a continuous function, we will see later, but these are never examples of these types. So we have 0 divided by power 4, constantly 0, it goes to 0.
59:04:970Paolo Guiotto: So does this contradict what we have done so far? No, we have two roads going to 0, 0, along which F has the same limit. We know that two roads are never sufficient to establish that the limit is something.
59:20:400Paolo Guiotto: Because we need to say, ALL roads.
59:23:790Paolo Guiotto: So two roads are not enough.
59:25:720Paolo Guiotto: So, we need to find a road, since the conclusion is already written, okay? Either there is something… a mistake in the assignment, or we have to look for this bad road. So, why don't we try to use the same roads we have seen here, straight lines.
59:45:310Paolo Guiotto: This is interesting because, in fact, this class, Y equals MX, for M equals 0 contains the x-axis. We cannot have the Y axis because there is no M for that case, but basically, it's a wide class of 9s, so maybe we are lucky we find something there.
00:02:350Paolo Guiotto: So let's see what happens if we do that calculation. So let's now take…
00:07:320Paolo Guiotto: a section like this, so this is the line Y equal MX, and this means that this point is the point XMX. So M is any real number, fixed, okay?
00:23:320Paolo Guiotto: What is variable is X. So, remind that this point goes to .00. If and or if.
00:32:870Paolo Guiotto: Even if X goes to 0, okay?
00:37:150Paolo Guiotto: Good. So let's see what happens when we evaluate F on these points. So we have…
00:43:70Paolo Guiotto: This is X times the square of MX.
00:47:630Paolo Guiotto: divided the X squared plus the square, no, the square, the power 4 of this.
00:53:730Paolo Guiotto: Okay, let's do the calculations. So we have an M squared X cubed divided by X squared plus m power 4, X power 4.
01:07:10Paolo Guiotto: Let's simplify. Remind that here, X is different from 0, otherwise we would be computing f at 0, so we have X different from 0. We can cancel something here.
01:19:410Paolo Guiotto: For example, you see that we can cancel, or maybe it's better to avoid a possible error. We write this. M squared, X cubed, we factorize X squared 1 plus m power, 4X squared.
01:34:650Paolo Guiotto: So we can now simplify this X squared, and it remains X.
01:39:950Paolo Guiotto: And nothing else can be done. So we have a final value, M.
01:45:810Paolo Guiotto: square x divided 1 plus m power for X squared. So, this time it is not constant.
01:53:130Paolo Guiotto: Now, what happens when x goes to 0, which is the condition to send that point to the origin?
01:59:290Paolo Guiotto: Now, if we send X to 0,
02:02:120Paolo Guiotto: This is a not particularly difficult limit, the numerator goes to 0, denominator goes to 1. So 0 over 1 equals 0.
02:13:360Paolo Guiotto: And this does not contradict what we have done so far, because it confirms that F is going to zero.
02:21:80Paolo Guiotto: We can say that along every straight line that goes to the origin, F goes to 0.
02:27:680Paolo Guiotto: Okay?
02:28:870Paolo Guiotto: So… Let's, emphasize this. Along… Each.
02:38:850Paolo Guiotto: Straight.
02:43:870Paolo Guiotto: lineup.
02:50:190Paolo Guiotto: going… to 0, 0, F goes to 0.
02:56:940Paolo Guiotto: Now, the question is, is that sufficient to conclude that F itself is going to zero?
03:10:10Paolo Guiotto: Yeah, that's right. We have seen this for straight lines.
03:14:950Paolo Guiotto: Maybe on some COVID line, things will go differently.
03:20:190Paolo Guiotto: Okay? And in fact, that's what happened.
03:23:80Paolo Guiotto: Now, to understand how we can find this strange line.
03:28:430Paolo Guiotto: Because this is the difficulty in doing this kind of job, searching the right sections. How can I search? Well, there are standard sections, straight lines, I can do calculations, but if I do, like, in this example, I see that I do not get any contradiction.
03:46:380Paolo Guiotto: So now the question is, how can I do? Well, let's go back a second to the function and see what… and let's give a look to the function to understand what can be done. You see that the function is made like that. XY squared divided by X squared plus Y power 4.
04:03:480Paolo Guiotto: It is clear that if the point XY goes to 0, 0,
04:08:340Paolo Guiotto: then both X and Y are going to 0.
04:13:380Paolo Guiotto: So this means that the numerator will go to 0, denominator will go to 0, we have this 0 over 0 in the terminated form.
04:22:430Paolo Guiotto: Now, if we have to think which among these two will be smaller? Yeah, it's a bit more complicated, because they are polynomials.
04:30:690Paolo Guiotto: But what is the degree of numerator and the degree of denominator?
04:35:500Paolo Guiotto: It seems that the numerator is like a third degree, no? Because X power 1, Y power 2, they are multiplied, so we know that if we have X times X squared is X cubed.
04:46:80Paolo Guiotto: So apparently, it seems to be a power… a third-degree stuff. So it's going to zero as the third power.
04:54:770Paolo Guiotto: About the denominator, it is not clear, because it seems a fourth degree.
05:00:40Paolo Guiotto: Because there is this Y with power 4.
05:03:460Paolo Guiotto: But, if you look at what happens.
05:07:310Paolo Guiotto: When we go to zero along the x-axis.
05:11:50Paolo Guiotto: Along the x-axis, this 0 to power 4, this term is just silent.
05:17:500Paolo Guiotto: So it is not a fourth degree, it's a second degree.
05:24:600Paolo Guiotto: So…
05:29:30Paolo Guiotto: Can we find a way to… To balance these two.
05:34:540Paolo Guiotto: Well, the way is… look what happens if I take…
05:39:780Paolo Guiotto: this, apparently strange section, which is, let's write, X equal Y squared.
05:52:600Paolo Guiotto: What is this?
05:55:50Paolo Guiotto: You know what cool is this in the plane XY?
06:00:400Paolo Guiotto: Yes, it is a parabola.
06:04:160Paolo Guiotto: Yeah, with the axis, the x-axis, this is made like that.
06:12:630Paolo Guiotto: This is the line X equals Y squared. You don't have to think, we are in the plane, there is no, preference between X and Y. They are exactly the same. So, this parabola is made like that, makes sense. So, here, points, it…
06:29:370Paolo Guiotto: It's better if we emphasize the Y, no? So, we write these points as Y squared Y,
06:37:870Paolo Guiotto: Because X equals Y squared.
06:40:80Paolo Guiotto: Now, what should happen to send this point to the origin 00, which is the case when Y is equal to 0?
06:48:700Paolo Guiotto: Now, this point, Y squared y goes to 0, 0, If and all if… Y goes to 0.
06:58:580Paolo Guiotto: Okay, let's see what happens if we evaluate F on these points. Y square, Y.
07:05:750Paolo Guiotto: You see that XY squared becomes Y squared times Y squared.
07:13:680Paolo Guiotto: Because you have just to replace X with Y squared.
07:16:930Paolo Guiotto: Then, downstairs, you have X squared, so which is Y squared to power 2 plus Y power 4.
07:25:510Paolo Guiotto: So, this means Y power 4 divided… Y squared squared means y power 4, plus another Y power 4 is 2Y power 4, and that's constantly equal to 1 half.
07:38:920Paolo Guiotto: You see? Therefore, when we send y to 0, that means we send this point to the point 0, 0,
07:47:660Paolo Guiotto: This quantity constantly goes to 1 half.
07:51:50Paolo Guiotto: And that's the way we want it, because now we have a road that goes to 0, 0, along which the function has a different limit.
08:01:430Paolo Guiotto: from the roads we found above. So, now we can conclude that there is no limit
08:10:530Paolo Guiotto: for XY going to 0, 0.
08:15:300Paolo Guiotto: of this F.
08:19:149Paolo Guiotto: Now, in general, it is not easy. There is not a standard recipe to find the right sections.
08:29:50Paolo Guiotto: And especially, imagine that this problem was posed in the form
08:35:00Paolo Guiotto: Look at this, discuss existence and value as the previous one.
08:40:620Paolo Guiotto: You don't know if the limit exists, so you have this alternative. We have not yet seen how to do to prove that the limit exists. We are just working on the limit does not exist.
08:53:120Paolo Guiotto: So, it's a bit complicated, because the first thing is you have… you need to… to make… to do, like, some guess. Is the limit…
09:01:140Paolo Guiotto: Going to exist.
09:03:170Paolo Guiotto: Do I think that the limit exists? So I should try to prove that. Or, if I think that the limit does not exist, my strategy is different. I should find different paths that go to the point here, 0, 0,
09:16:720Paolo Guiotto: where the limit is different. How to find these paths? Of course, I can try with standard paths, maybe I'm lucky with… as in the case we have here.
09:26:210Paolo Guiotto: Simple sections, simple straight sections are simple sections to do calculations, and you can… if they are good, like in this case, you are done. But what if they are not sufficient?
09:41:439Paolo Guiotto: So, the first message is that even if I have that along all straight lines the limit is the same, this does not mean at all that the limit is that value.
09:53:50Paolo Guiotto: Maybe the limit still does not exist, and because I can find another section. Now, how do you find these kind of sections? Well, here, unfortunately, there is not an algorithm, a rule, I can tell you you have to do this or that.
10:09:760Paolo Guiotto: Here, you have to be creative. You have to try to understand how to balance the quantities to make, for example, a quantity different from zero. Constant different from zero.
10:20:630Paolo Guiotto: So here, it's not easy, but maybe by looking at that, you can see that you can make the two powers downstairs the same thing. If you balance the exponent, you could put Y equal the square root of X, for example, which is exactly the same line, it's just the…
10:39:560Paolo Guiotto: Positive branch of that line, huh?
10:41:920Paolo Guiotto: If you put Y equal root of X, you get an X square downstairs, so it's a second degree, and upstairs, you have X times root of X squared, which is another X, so that's X square. So you see, second degree, second degree, you get something different from zero, because they balance perfectly. Okay?
11:03:60Paolo Guiotto: There is an exercise on this.
11:07:00Paolo Guiotto: At the end of the chapter.
11:13:550Paolo Guiotto: Which is the exercise, 187. So, I warmly suggest to do this, by tomorrow.
11:24:60Paolo Guiotto: Do we have cast tomorrow? Yes.
11:27:30Paolo Guiotto: So, do… Exercise them.
11:32:100Paolo Guiotto: 187. These are easy, so you should understand
11:37:590Paolo Guiotto: it's an exercise on disproving existence. Perhaps there is one of these which is a little bit more difficult because it's a limit in three variables, but all the others are in two variables, and
11:51:470Paolo Guiotto: It should be easy to understand which sections should help you to find out the answer, okay? So, it's something that I cannot codify in any way, so you have to try to get a little bit of feeling with these problems, and then tomorrow morning, we will see.
12:09:800Paolo Guiotto: Some solutions of these problems.
12:14:640Paolo Guiotto: Okay.
12:17:130Paolo Guiotto: So…
12:18:820Paolo Guiotto: I want now to show an example, because so far we have seen examples of limits that do not exist, so the strategy is clear. You have to find two sections along which F has two different limits.
12:33:510Paolo Guiotto: Now, let's see this other example, which is still a variation on the same kind of…
12:40:810Paolo Guiotto: problems. This is… we may say this is a 2-star.
12:45:180Paolo Guiotto: As difficulties. So, pretty standard.
12:48:650Paolo Guiotto: Limiter for XY.
12:52:150Paolo Guiotto: Going to 0, 0.
12:55:910Paolo Guiotto: Here we have XY squared, divided X squared plus Y squared.
13:03:90Paolo Guiotto: So it's similar to the previous one. All these problems are very similar, we change little things. For the moment, we do not put more complicated functions like exponentials, we are just working with polynomials, which are simple.
13:17:660Paolo Guiotto: Okay, so, first of all, the function here is, FXY equal XY squared divided X squared plus Y squared, defined…
13:31:610Paolo Guiotto: on D, the domain, as you can see, it is the set of points XY in R2.
13:40:560Paolo Guiotto: Such that X squared plus Y squared, the denominator, is different from 0.
13:47:700Paolo Guiotto: Now, this is possible only if the point is the .00, so you have everything except .00, which is exactly the point where we have to compute that limit, okay?
14:04:260Paolo Guiotto: So, this is the… the picture…
14:08:560Paolo Guiotto: the plane XY, the point 0, 0, where we have to compute the limit, and the domain D, which is everything except that single point. So, it is clear that D surrounds the point 0, and so clearly.
14:25:960Paolo Guiotto: The .00 is an accumulation point for the domain, so the limit makes sense.
14:34:380Paolo Guiotto: Okay, so what is now the first thing to do? Let's see, with some sections, what happens. Maybe we get some filling.
14:43:60Paolo Guiotto: So, if I use the standard axis sections, for example, the x-axis, I evaluate on point 60, with X different from 0.
14:55:20Paolo Guiotto: This is X times 0 divided 0 square, divided X squared plus 0 square. So again, 0 divided X squared, constantly equal to 0, goes to 0.
15:08:960Paolo Guiotto: As you can imagine, there is not a big difference with the y-axis. We get also, for the y-axis, 0Y.
15:18:470Paolo Guiotto: Constantly 0, it goes to zero. This when x goes to zero, while this when Y goes to 0.
15:27:200Paolo Guiotto: Now, at this stage, what can I say?
15:31:220Paolo Guiotto: I have two sections of F.
15:34:50Paolo Guiotto: That have the same limit, zero.
15:37:550Paolo Guiotto: So this is not at all saying that the limit exists.
15:41:640Paolo Guiotto: And, of course, it does not say that the limit does not exist.
15:45:220Paolo Guiotto: The only thing that can say… I can say at this stage is that if a limit exists, it must be zero.
15:52:260Paolo Guiotto: Because if a limit exists along each section, F goes to the same value. If I have 1 or 2 or 10 sections along which I go to the same value, 0,
16:05:740Paolo Guiotto: The unique possibility for limit is to be zero. There is no other possibility. You understand this? So at this stage, I can say…
16:20:560Paolo Guiotto: I would… Say… That, if… a limit.
16:32:120Paolo Guiotto: of F, when XY Goes to 0, 0.
16:37:110Paolo Guiotto: exists.
16:40:720Paolo Guiotto: then… Eat.
16:42:930Paolo Guiotto: Must.
16:45:180Paolo Guiotto: B.
16:47:380Paolo Guiotto: So, in other words, it's not a completely useless work, this one of computing some sections, because it provides a candidate for the limit. There is a unique possible limit, zero.
17:03:30Paolo Guiotto: So we already know that if there is a limiter, we have the value of the limit.
17:07:500Paolo Guiotto: So now the road, we have a bifurcation. So we have to decide if to try to disprove the existence, and then we have to find another section along which F does not go to 0.
17:22:160Paolo Guiotto: Or, if we want to try to prove that the limit is 0, because there is no other possibility.
17:29:420Paolo Guiotto: Now, here there is a good reason to, to say that probably this limit should be zero. Why? Because if we look at that function, we see that it is a ratio of polynomials. At numerator, we have something like third degree.
17:47:350Paolo Guiotto: Okay? At the denominator, we have a second degree power, so it's definitely a second degree. So, if I have one… only one variable, X cubed divided X squared, the ratio is X, the limit is 0.
18:01:930Paolo Guiotto: But here, unfortunately, I have not one variable, I cannot simplify anything here. So I have now to find a way to make this intuition a precise argument.
18:14:420Paolo Guiotto: Now, the trick here is actually a standard trick, which is the following. So, our guess is…
18:23:620Paolo Guiotto: There exists the limit, for XY going to 0, 0,
18:30:180Paolo Guiotto: of our F, and this limit is 0. This guess is based on the rough argument I have done by words right now. But now I will make a precise argument. And the precise argument is based on this idea… well, we know that points in plane
18:45:910Paolo Guiotto: can be represented in the natural way with the Cartesian coordinates, but there is another, and here a very important way to characterize a point in plane, which is using polar coordinates.
19:01:200Paolo Guiotto: Polar coordinates are two numbers. In this case, one number is the distance to the origin that we will denote with the letter rho, and the other is the angle made
19:13:490Paolo Guiotto: the agreement is with the positive direction of the x-axis. So this theta is a value between 0 and 2 pi.
19:23:780Paolo Guiotto: So this means that X is rho cosine theta, Y is raw sine theta. So this is the relation between X and Y and rho n theta, no?
19:37:540Paolo Guiotto: And the important point is that what is rho is an important quantity for our problem, because rho is the distance of point XY to the origin. So, literally, it is the square root of X squared plus Y squared.
19:52:900Paolo Guiotto: And this quantity is what?
19:55:110Paolo Guiotto: The norm of vector XY.
19:57:900Paolo Guiotto: And why this quantity is important?
20:00:700Paolo Guiotto: Because we are computing limit at XY going to 0.
20:05:740Paolo Guiotto: So, think about, XY goes to 0, yes, if and only if X and Y goes to 0. But, also, if and only if, the distance between X and Y and the vector 0 goes to 0.
20:21:10Paolo Guiotto: That's actually the definition, no?
20:24:90Paolo Guiotto: But the norm of XY minus 0 is the norm of XY.
20:32:470Paolo Guiotto: So this must go to zero, and this is nothing but raw.
20:38:480Paolo Guiotto: So, we have that the point goes to zero if and only if raw goes to zero.
20:47:240Paolo Guiotto: Now, our time is almost over, so let's see if we can, in a couple of minutes, complete this.
20:53:930Paolo Guiotto: So this is important because if we take the function f.
20:57:970Paolo Guiotto: And now, write this by using the polar coordinates, so this means you replace X with the raw cosine theta. You replace Y with the raw sine theta, what you get is… so the function is XY square.
21:11:330Paolo Guiotto: Divided X squared plus Y squared. Okay, let me just copy here. XY square divided X squared plus Y squared.
21:19:180Paolo Guiotto: Now, with polar coordinates, We have that X is raw cos theta.
21:28:110Paolo Guiotto: Then we have Y squared is raw square sine squared theta, divided by X squared plus Y squared
21:37:620Paolo Guiotto: It's not one.
21:40:130Paolo Guiotto: Raw?
21:42:20Paolo Guiotto: Raw square, exact.
21:44:460Paolo Guiotto: Now, look at what happens with the raw and the theta. We can separate the two. We get a raw cube divided the raw square times cos theta sine squared theta.
21:57:40Paolo Guiotto: Now, if you look at this part of the story, that's exactly cube divided square I told you before.
22:04:430Paolo Guiotto: And here you can simplify, so you can say that this is algebraically equal to this, so you get that our F, XY is raw times quantity, which is this box here, cos theta sine squared theta.
22:23:380Paolo Guiotto: That… the intuition is easy now, because we know that, remind, point XY goes to 0, 0 if and all if raw goes to 0. So in this expression, we see that we have a quantity which is going to zero.
22:35:890Paolo Guiotto: This quantity is… what is doing this? We don't know, but the key point is that being a product of sine and cosine is a quantity which is not so big.
22:46:850Paolo Guiotto: It is actually bounded, because sine and cosine oscillates between minus 1 and 1.
22:52:410Paolo Guiotto: So this number cannot be larger than 1 and smaller than minus 1. So, if this is going to 0, this is bounded, this quantity will go to 0. How can we formally see this?
23:05:80Paolo Guiotto: So take the absolute value of this.
23:08:620Paolo Guiotto: You have that this is the absolute value of this quantity.
23:14:190Paolo Guiotto: Which is the absolute value of raw. And this is raw because raw is positive, raw is the distance.
23:20:570Paolo Guiotto: Times the absolute value of cosine theta, times the absolute value of sine squared theta.
23:27:550Paolo Guiotto: These two numbers are both less or equal 1,
23:32:690Paolo Guiotto: So this means that the modulus of FXY
23:37:850Paolo Guiotto: is less or equal, because it is equal to rho times something which is less or equal than 1, it is less or equal than rho.
23:45:550Paolo Guiotto: And since this goes to 0, when rho goes to 0, I remind you that this is equivalent of saying XY goes to 0, 0.
23:56:150Paolo Guiotto: automatically, the modulus of F must go to 0 because of the argument of the squeeze theorem, okay? So this means that F must go to 0 by squeeze
24:12:860Paolo Guiotto: Theorem.
24:14:930Paolo Guiotto: And therefore, we have the conclusion.
24:17:10Paolo Guiotto: Now, I… I closed it with the…
24:19:950Paolo Guiotto: fast, because I have to finish, but we will return on this argument tomorrow.
24:26:40Paolo Guiotto: So, what I ask you is, do the exercise I left on sections, so that's disproving existence of limit, and review carefully this example. Tomorrow, I will return on this, and we will see more examples.