AI Assistant
Transcript
00:10:450Paolo Guiotto: I left you to do some exercise, wow.
00:19:00Paolo Guiotto: In particular, the exercise, one… 8… Seven.
00:28:630Paolo Guiotto: So, check.
00:32:890Paolo Guiotto: But…
00:37:440Paolo Guiotto: on the roof.
00:41:140Paolo Guiotto: Limits.
00:45:230Paolo Guiotto: Not.
00:47:610Paolo Guiotto: insists.
00:49:450Paolo Guiotto: Let's say the number 1, which is the limiter for XY, Going to zero.
00:57:790Paolo Guiotto: in the plane.
01:00:330Paolo Guiotto: of X squared minus Y squared divided by X squared plus Y squared.
01:12:290Paolo Guiotto: Okay, so here, the function f is…
01:16:740Paolo Guiotto: X squared minus Y squared divided X squared plus Y squared.
01:23:490Paolo Guiotto: It is defined…
01:28:520Paolo Guiotto: The natural domain, which is, clearly everything except .00.
01:38:210Paolo Guiotto: And, clearly, 00 is an accumulation point for the domain.
01:47:780Paolo Guiotto: So, the limit, makes sense
01:52:300Paolo Guiotto: Now the question is, we have to check that
01:55:860Paolo Guiotto: the limit does not exist. And to do this, we have to find at least two different ways to go to 0, along which the function has two different behaviors.
02:06:470Paolo Guiotto: to different limits.
02:08:479Paolo Guiotto: Have you done this?
02:11:70Paolo Guiotto: So, what should I compute?
02:15:110Paolo Guiotto: You.
02:18:310Paolo Guiotto: Goodbye.
02:22:60Paolo Guiotto: This is zero.
02:23:210Paolo Guiotto: Yeah, so you say F0?
02:28:210Paolo Guiotto: Zero?
02:30:90Paolo Guiotto: No. What? Why most people?
02:33:10Paolo Guiotto: Y equals zero, so X0.
02:37:140Paolo Guiotto: this.
02:39:260Paolo Guiotto: Okay, so if we have this, we put Y equals 0, we get X squared divided by X squared, which is constant equal to 1, it goes to 1 when, so the point X0 goes to 0, so it goes to the origin if and only if X goes to 0.
02:59:60Paolo Guiotto: So this goes to 0, 0.
03:02:260Paolo Guiotto: If, and only if X goes to 0.
03:06:520Paolo Guiotto: So, along this, the limit is 1.
03:09:20Paolo Guiotto: And then…
03:13:770Paolo Guiotto: F0Y.
03:17:640Paolo Guiotto: Okay, yes, because if you do that, you get minus Y squared.
03:23:780Paolo Guiotto: divided Y squared, so it feels constant equal to minus 1. It goes to minus 1, so now we are here.
03:33:730Paolo Guiotto: when Y goes to zero. So we have two different limits.
03:38:370Paolo Guiotto: And this is, enough.
03:40:940Paolo Guiotto: Okay, so… From this, there is none.
03:44:760Paolo Guiotto: the limit.
03:46:700Paolo Guiotto: for X, Y, Going to 0, 0.
03:51:110Paolo Guiotto: I'll bet.
03:54:10Paolo Guiotto: Okay.
03:54:950Paolo Guiotto: Number… Number 2?
04:00:720Paolo Guiotto: So yeah, these are easy, XY going to 0, 0.
04:09:170Paolo Guiotto: of, X square… plus Y cubed divided by by X squared… plus Y squared.
04:22:140Paolo Guiotto: Okay, so again, here we have, F… Which is this one.
04:32:790Paolo Guiotto: is defined… on domain D, which is everything, as you… as in the previous examples, except 00.
04:47:400Paolo Guiotto: And, share what the… Can we do?
04:54:630Paolo Guiotto: Same sections.
04:56:460Paolo Guiotto: Faccido… In this case, we get X squared over X squared, so constantly 1, it goes to 1.
05:04:940Paolo Guiotto: when X goes to 0, If we do F0Y, we get,
05:11:200Paolo Guiotto: Y cubed divided by Y squared, so it is Y.
05:16:390Paolo Guiotto: that now goes to 0 when y goes to 0. So in any case, we have two different limits and a limit.
05:25:100Paolo Guiotto: does not exist.
05:33:860Paolo Guiotto: Okay.
05:36:750Paolo Guiotto: Is there any of these that you won't particularly see?
05:44:810Paolo Guiotto: Well, let's take, for example, number 4, which is a limit with, with the three variables.
05:55:350Paolo Guiotto: We have the limit.
05:58:180Paolo Guiotto: for XYZ, that goes to 000.
06:04:170Paolo Guiotto: of X plus Y squared… Las…
06:11:360Paolo Guiotto: Z cubed divided by the root off.
06:16:750Paolo Guiotto: X squared plus Y squared plus Z squared.
06:23:330Paolo Guiotto: Okay?
06:24:760Paolo Guiotto: So, yeah.
06:27:900Paolo Guiotto: Yeah?
06:29:670Paolo Guiotto: Thank you.
06:31:440Paolo Guiotto: Yeah, one sec.
06:36:10Paolo Guiotto: Here, F, X, Y, Z.
06:39:770Paolo Guiotto: is this function.
06:42:690Paolo Guiotto: It is, is defined…
06:46:690Paolo Guiotto: This time we have a domain of R3, so D would be the set of arrays, XYZ, of R3, such that you see that the numerator is always defined.
07:01:790Paolo Guiotto: also the root at the denominator, because X squared plus Y squared plus Z squared is definitely positive.
07:11:180Paolo Guiotto: So the root is well-defined, and what could be a problem is that the root can be zero.
07:16:280Paolo Guiotto: And it can be 0 when the argument of the root is 0. So we need that quantity X squared plus Y squared plus Z squared be different from zero.
07:28:140Paolo Guiotto: Now, that's equal to zero, since it's the usual sum of positive things equals zero, if and all if each of the three terms are 0. So this is, once again, the full, in this case, space at 3, except .000.
07:48:700Paolo Guiotto: So if we want to do a figure, here we are.
07:51:650Paolo Guiotto: In R3, so we have 3 axes.
07:57:360Paolo Guiotto: So there are a lot of more complicated sections we can do. However.
08:03:410Paolo Guiotto: We could do just the simplest, so moving along axis, let's see what happens here. Along the x-axis, we have points of type X00.
08:15:50Paolo Guiotto: So, if you evaluate these points, we have… This. Numerator becomes axon.
08:24:170Paolo Guiotto: denominator becomes root of X squared.
08:29:10Paolo Guiotto: Which is?
08:31:760Paolo Guiotto: Modulus of X, exactly.
08:37:10Paolo Guiotto: You know what is the value of this quantity X divided modulus of X?
08:44:700Paolo Guiotto: If X is positive, this is positive.
08:49:870Paolo Guiotto: One, if X is negative?
08:52:540Paolo Guiotto: And that quantity is the… sine of X.
08:59:160Paolo Guiotto: Well, we here don't need to find other sections, because we could see that when X goes to zero, the limit of sine does not exist, because this is positive or negative. So we could say that if we move the point along the positive direction.
09:16:300Paolo Guiotto: So if we take FX00 with X going to 0, positive.
09:23:670Paolo Guiotto: On the right side of the axis.
09:27:10Paolo Guiotto: This quantity will be constantly equal to 1, so it goes to plus 1.
09:33:930Paolo Guiotto: While, if we go to zero from the negative part, so it means that if we take points down here.
09:41:950Paolo Guiotto: X 00, and we move to the origin that way, no? So these are points going to zero with X going to zero negative, and these are points going to zero with X going to zero positive.
09:56:00Paolo Guiotto: In the second case, the sine is equal to minus 1, so we have that the function is constantly equal to minus 1, the limit will be minus 1. So here we have two ways to go to 0,
10:09:860Paolo Guiotto: It doesn't necessarily mean that you have to put all the inferior line, okay? You go to zero. From one side, you get plus 1, you go to 0. From the other side, you get minus 1. These are two ways along which they go to 0, 0, but the function goes somewhere else, so to two different values.
10:29:40Paolo Guiotto: So this is sufficient to conclude that there is no limit.
10:35:570Paolo Guiotto: for F at 000.
10:38:410Paolo Guiotto: Is that clear?
10:41:520Paolo Guiotto: Okay, so these are two roads.
10:44:300Paolo Guiotto: Going to 000.
10:50:830Paolo Guiotto: Maybe let's give a look to the number 6, which has 2 stars.
10:55:130Paolo Guiotto: I don't know why.
10:57:260Paolo Guiotto: But perhaps there is some difficulty.
11:00:90Paolo Guiotto: We have a limit still in 3 variables.
11:05:560Paolo Guiotto: XYZ going to 0, 0, 0.
11:11:120Paolo Guiotto: Our function is XZ divided by… X power 4 plus 10.
11:19:180Paolo Guiotto: Y power 2 plus Z power 2.
11:25:100Paolo Guiotto: Okay?
11:26:450Paolo Guiotto: Now, also here, this is the function F. F is defined on…
11:39:140Paolo Guiotto: the domain D is what? Everything, because the numerator is always defined, whatever X, Y, Z are. The denominator, same thing. The unique problem is that when denominator is zero, you cannot do the division. So, you should say points X, Y, and Z such that
11:56:920Paolo Guiotto: that quantity X power 4 plus Y squared plus Z squared is different from 0.
12:03:920Paolo Guiotto: Now, that quantity is zero, we are again with the sum of positive things, so the sum is zero, even though if all terms are zero. So this means that we have everything except the unique possibility
12:18:650Paolo Guiotto: For this quantity to be 0 is the point 000.
12:23:540Paolo Guiotto: Of course, 000 is an accumulation point for this domain, because domain is everything except that point, so the limit makes sense, and now we can see what is the behavior of this thing along sections.
12:40:340Paolo Guiotto: Now, what kind of section could we take here?
12:45:620Paolo Guiotto: And normally, I told you, there is not a particular strategy, let's say, there is not a standard strategy to find out what are the right sections, but you could start with the axis, because there are many zeros, and this implies, that's the reason, not the special reason.
13:04:260Paolo Guiotto: So if I take X00, for example.
13:07:840Paolo Guiotto: This gives 0 divided by X power 4. Remind that X is different from 0 here, otherwise the point would be 000, which is never a load, the limit point. So this fraction is well-defined, and it gives 0 constantly. It goes to 0 when x goes to 0.
13:28:70Paolo Guiotto: Now, if you, you should take the other axis, no?
13:35:10Paolo Guiotto: You see that when x is 0, numerator is 0. When Y is 0, Well,
13:46:880Paolo Guiotto: So, you have always two coordinates, sorry. So, if you have two coordinates, at least one of X and Z is 0, so we do not change too much with the other axis. So, now, if we want to find something different, we need to find something different from the other two axes.
14:03:660Paolo Guiotto: So what could be the idea?
14:10:990Paolo Guiotto: X and Z, what?
14:14:810Paolo Guiotto: Yes, but, I mean, if you say this, F of X0Z, is that?
14:22:830Paolo Guiotto: Yeah, but the point is that by doing this, you have still a function of two variables, because you get XZ divided
14:33:380Paolo Guiotto: That's for… plus Z squared.
14:37:200Paolo Guiotto: So it is not yet so clear what happens.
14:40:810Paolo Guiotto: is equal to X squared.
14:44:960Paolo Guiotto: you say Z equal X squared.
14:49:780Paolo Guiotto: Okay, let's see. So you say F, X,
14:53:950Paolo Guiotto: 0 and X squared, is that…
14:57:360Paolo Guiotto: So if we do that, we have X times X squared divided X power 4 plus Z squared means X power 4. So it gives X cubed divided 2, X power 4,
15:10:390Paolo Guiotto: So it is 1 half times 1 over X. It seems good, because… now, it's not immediate to plot this, even if this is actually an easy curve.
15:22:690Paolo Guiotto: Because in the space XYZ, if we want…
15:27:00Paolo Guiotto: What is this? This is Y equals 0.
15:30:780Paolo Guiotto: means that we are on the plane XZ, no? The point where the Y is called negative zero means that for Y, we are here, so we move on the plane XZ.
15:42:760Paolo Guiotto: On the plane X, we see that we have X and Z equal X squared, so that's a parabola contained into this plane. So if you want to see the curve, it is Z equal X squared, something like…
15:55:680Paolo Guiotto: Fun.
15:58:280Paolo Guiotto: I'm sorry, I'm pretty sure that you do not understand Italian, so do not be my Italian.
16:06:450Paolo Guiotto: I could be…
16:10:910Paolo Guiotto: So this curve is a parabola Z equals X squared and Y equals 0. It's a parabola in space contained in the plane XZ.
16:22:870Paolo Guiotto: However.
16:24:690Paolo Guiotto: It is clear that the point X0, X squared goes to 000, which is our limit point, if and only if
16:37:920Paolo Guiotto: X goes to 0, right.
16:39:990Paolo Guiotto: So what happens to 1 over X when x goes to 0?
16:46:200Paolo Guiotto: It goes to plus minus V, depending on…
16:51:920Paolo Guiotto: If you go to zero from the right to left. No, so this can go to plus infinity, if you go X, go to 0 from the right, this minus infinity if you go from the left. Okay, I'm sorry, from the right and from the left.
17:07:800Paolo Guiotto: And even this would be sufficient, because you see.
17:12:260Paolo Guiotto: It means that if you go to zero, from the right is this part of the parabola. On this part, you have this behavior of F.
17:22:780Paolo Guiotto: So, when your point is here, X0X squared.
17:27:650Paolo Guiotto: moves to the origin along the red part of the parabola, the function goes to plus infinity.
17:34:560Paolo Guiotto: Okay? If you evaluate the function on the green part of the parabola, these are the same points, X0, X squared, but now the red points are with X positive, the green points are with X negative.
17:49:560Paolo Guiotto: On these green points, the function goes to minus infinity.
17:55:470Paolo Guiotto: So we are.
17:56:930Paolo Guiotto: We are going to the same point, 000, but the function is going to two different values.
18:02:760Paolo Guiotto: So this means that there is no limit.
18:05:320Paolo Guiotto: Even more, we had already checked that this, moving along the x-axis, we go to zero, so definitely there is no limit.
18:23:350Paolo Guiotto: Any questions?
18:27:410Paolo Guiotto: Also, use, opinion.
18:32:440Paolo Guiotto: Thank you.
18:33:700Paolo Guiotto: Yes, I don't know.
18:39:300Paolo Guiotto: Y-yeah.
18:42:780Paolo Guiotto: Yes, yes, you just need to find the tour. You don't have to find
18:48:530Paolo Guiotto: 1,000, okay. But it's, it's clear that there are many choices. You suggest to do this, is that correct?
18:57:960Paolo Guiotto: Well, if you do this calculation here, you would get X times Z, so MX squared divided by X power… was X power 4?
19:08:850Paolo Guiotto: Yes, Y squared, Y0, Z squared, so plus M square, X squared. Here we should use, of course, M different from 0.
19:19:690Paolo Guiotto: M equals 0 is the x-axis. For M different from 0, we see that we factorize x squared in the denominator, we get this m square plus X squared. We simplify this X squared, and finally we compute the limit.
19:38:640Paolo Guiotto: Now, the point X0, MX goes to 0 if and or if X goes to 0, so we send X to 0 and we get M divided M squared, so 1 over M.
19:52:500Paolo Guiotto: which is non-zero for M different from zero, and for different values of M, you get a different value, so…
19:59:850Paolo Guiotto: It's, it's, oh, it's good as well.
20:03:390Paolo Guiotto: Okay.
20:05:510Paolo Guiotto: Okay, so let's say that we… Are now a beta.
20:12:200Paolo Guiotto: Confident with this, problem of sectioning a function.
20:17:120Paolo Guiotto: to… look for…
20:20:510Paolo Guiotto: bad sections to disprove limits, at least in some cases. Then we relax, then we generalize.
20:29:870Paolo Guiotto: Okay, now let's focus on the problem of showing existence of limit. We have done an example yesterday. I want to do an example right now to retake these ideas we have seen yesterday, and then we try to
20:45:890Paolo Guiotto: find out a sort of general strategy that could be applied in different circumstances. So I will do that by doing some of the exercises 188. So I…
21:02:730Paolo Guiotto: Now, do some of them, you finish with the other that I leave.
21:08:300Paolo Guiotto: Now, so let's start with the example one, which is a one-star, so easy.
21:14:210Paolo Guiotto: Easy does not mean that, mean easy. Mean that this, it's the wrong word, actually, easy. One star means everybody here must understand.
21:26:490Paolo Guiotto: To start, the most of you should understand, okay?
21:30:20Paolo Guiotto: So, let's say that one study is the minimal, it's not enough, but it's something that you must, you must, you must be, you must be confident with that thing.
21:42:530Paolo Guiotto: But you must reach the 2 star, because 2 star means this, this is the expected level. Then, whatever is above this is, is something…
21:53:140Paolo Guiotto: that's it.
21:56:240Paolo Guiotto: You must, that it should be the summer.
22:04:230Paolo Guiotto: work. Okay, so this is the limit when XY goes to 0 of X times Y divided by the square root of X squared plus Y squared.
22:15:780Paolo Guiotto: So, again, the exercise… well, the exercise says compute the following limits, but let's paste each of these exercises as if we don't know, of course, if the limit exists or not. So, we have a limit, and let's try to see what happens.
22:30:250Paolo Guiotto: So, here, we have our function f, which is, of course, this thing I do not rewrite.
22:38:270Paolo Guiotto: And, F is defined
22:42:930Paolo Guiotto: on domain D, which is, as usual, as you can see, also here, both numerator and denominator are well-defined, whatever are X and Y.
22:54:120Paolo Guiotto: then you have a fraction, and you cannot have the denominator equal to zero. So you need that the root be different from zero. The root is 0 when the argument is zero. So the unique problem is,
23:07:720Paolo Guiotto: These are all XY, except,
23:11:590Paolo Guiotto: those for which X square plus y squared is equal to 0.
23:15:380Paolo Guiotto: And now, this is the unit 0.00, so again, add to Minus.
23:22:720Paolo Guiotto: 0, 0.
23:24:800Paolo Guiotto: So we don't need to say any time 00 is clearly an accumulation point, so the limit makes sense, blah blah blah.
23:31:640Paolo Guiotto: So let's now focus on this.
23:35:710Paolo Guiotto: Okay, so the first thing… this is a general rule. Whenever you have a limit, the first thing to do is to recognize if there is any issue. This means any indeterminate fall.
23:46:670Paolo Guiotto: What can be said here? X and Y go to 0, 0. So, this means that both X and Y are going to 0, and this means that the product of the numerator goes to zero, the argument of the root will go to 0, so the root will go to 0, and therefore we have a 0 versus 0.
24:05:570Paolo Guiotto: Now, in this battle, we have to understand if one of them is stronger than the other. Stronger means here, because they are small, which one is bigger?
24:16:420Paolo Guiotto: less small than the other. Or if they are the same size, so in this case, there is not one of them that kills the other, and that's… the limit won't be zero, probably, or infinite.
24:30:730Paolo Guiotto: Now, what can be said about this?
24:33:550Paolo Guiotto: Let's try to attack this with the same strategy we used yesterday. Yesterday, we introduced this idea that instead of using Cartesian coordinates, which are a normal way
24:45:980Paolo Guiotto: We have limits, functions. We use polar coordinates, another couple of variables, which are a distance, draw, and an angle.
24:58:120Paolo Guiotto: This is a very important, say, change of variable, because, while, X and Y
25:07:730Paolo Guiotto: goes to 0, 0, means,
25:10:980Paolo Guiotto: both X and Y are going to zero, but you don't know exactly how they are infinity in many ways, okay? So they are independent, in some sense.
25:22:580Paolo Guiotto: For the Cartesian coordinates, this boils down to rho goes to zero, and now what about theta?
25:32:280Paolo Guiotto: Well, about theta, you cannot do anything, because… look, imagine that your point is moving to zero in this way, to a long straight line. So, raw goes to zero, and theta remains constant.
25:47:960Paolo Guiotto: Okay, so theta is not even moving.
25:50:890Paolo Guiotto: is not going to zero, to be clear. It's not… it's staying there to a certain value. Imagine that you go to zero along with something like a spiral, like this.
26:02:90Paolo Guiotto: So, the raw, which is this distance, as you can see, shrinks down to zero.
26:07:730Paolo Guiotto: And theta changes 0 to 2 pi, then restarts from 0 to pi, 0 to pi, forever.
26:14:790Paolo Guiotto: So, the meaning… the fact is that the unique thing you can say on theta is that theta is bounded between 0 and 2 pi, so there is no other information. So, in a sense, you see that there is a… while in XY coordinates, there is a sort of symmetry.
26:33:620Paolo Guiotto: the two coordinates, the point, XY goes to 0, 0, even or if the two coordinates go to 0, so they are the same. It doesn't matter who is the absister, the ordinate, they must go both to zero. With polar coordinates, there is an asymmetry.
26:47:610Paolo Guiotto: Because one of the two goes to zero, the other does not go anywhere, so it's just bounded in intervals 0 to pi.
26:56:60Paolo Guiotto: This is a key point because it's going to transform a limit in several variables into a limit in one variable, which is something we know how to deal with.
27:07:420Paolo Guiotto: Because you learned to compute limits in one variable, no?
27:11:950Paolo Guiotto: Okay, so now, let's see what happens if we just replace the polar coordinates into the function f. So, if we write FXY,
27:21:210Paolo Guiotto: in polar coordinates, so X equal raw cosine theta, and Y equal raw sine theta. This is, let's say, the relation between XY raw theta.
27:35:140Paolo Guiotto: This is the relation that expresses XY functions of raw theta, okay? There is also an opposite relation, but we don't care here for the moment.
27:46:710Paolo Guiotto: Now, in this case, we have… the function was X times Y, so I have raw cosine theta.
27:54:430Paolo Guiotto: that's X times raw sine theta, that's Y, divided… here, we are lucky, because the root of X squared plus Y squared
28:03:780Paolo Guiotto: is exactly what means raw. Raw is the root of X squared plus Y squared. It is the Euclidean distance to the origin. So I can say that, if you want, you replace raw cos theta raw sine theta, you do the sum of squares, and you remind that cos squared plus sine squared is 1.
28:22:610Paolo Guiotto: So you get raw.
28:25:420Paolo Guiotto: Now, you see that here, first of all, we can simplify something, so let's do that, because we have a raw square divided raw, and then we have a factor, cos theta sine theta.
28:39:290Paolo Guiotto: Now, as you can see, let's remain first before we do the communication. We remain the second one here.
28:45:20Paolo Guiotto: Why didn't you see the reason why you have 0 divided 0?
28:48:750Paolo Guiotto: who is the responsible of these zeros. It is this role.
28:53:80Paolo Guiotto: It is certainly not cosine theta or sine theta. They are zero only for spatial values of theta.
28:59:440Paolo Guiotto: But we say that theta could be a constant where these two are not zero. So, we cannot think that the fact that numerator goes to zero depends on some value of theta. No, it depends on the fact that there is this factor rho, actually raw square up, that goes to zero.
29:15:250Paolo Guiotto: In the realm, there is not even Tita, there is just rock.
29:18:860Paolo Guiotto: And here, you see that the numerator, we have a raw times raw, so it's about the raw square.
29:24:920Paolo Guiotto: So, it's an order 2, 0 of order 2. And here we have a first degree rule.
29:32:210Paolo Guiotto: Now, we know that powers, when the exponent is higher, they go faster to zero when the argument goes to zero.
29:40:500Paolo Guiotto: So this quantity is definitely smaller than this one. Now, think about raw equals 1 over 10, raw square is 1 over 100.
29:50:570Paolo Guiotto: raw equals 1 over 100. This is 1 over, 10,000, right?
29:57:720Paolo Guiotto: So you see that raw square is definitely much smaller than raw, so that's its place, why if I have to bet on something here, it's not computing any here. This is just F is equal, is F. This is F, but written it with other coordinates.
30:14:320Paolo Guiotto: If I have a look at this expression, I should say, before, to simplify things, I should say, the numerator is smaller order of denominator, so this probably will go to zero.
30:25:990Paolo Guiotto: And now I get, okay, the point. I check that limit is zero, so it's completely a waste of time looking for sections on which I have to limit, you see? Because the bet is now, definitely, the limit exists, and it is equal to zero.
30:43:520Paolo Guiotto: Now, the problem is just this.
30:46:720Paolo Guiotto: Screen. Okay.
30:49:140Paolo Guiotto: So, let's simplify, and let's verify this. So here, we see that we simplify this two, and several times, this quantity
30:58:240Paolo Guiotto: this is going to zero, this is not going anywhere, but it is bounded, so it does not bother too much. Now, I… how can I do this, a formal argument? I can notice that if I take a modulus f of XY,
31:14:270Paolo Guiotto: But before writing modules, say, I bet that F is going to 0, right?
31:19:680Paolo Guiotto: So I say, take the difference between F and the candidate limit, no?
31:26:100Paolo Guiotto: That absolute value is going to assess how much F is different from 0, which should be the limit. I will expect that this quantity will go to 0 when XY go to 0. In fact, this quantity is the absolute value of FXY, because subtracting 0 is…
31:45:860Paolo Guiotto: subtracting nothing, then I have models of rho cosine theta, sine theta.
31:52:680Paolo Guiotto: Now, modulus of the product is the product of the modulus. Modulus of raw is raw, because raw is positive, okay? Always, remain in touch with the…
32:04:360Paolo Guiotto: The problem, so in particular, raw is what? The raw is the distance to the origin, so it's not any number, it's positive.
32:13:520Paolo Guiotto: So raw, and then we have models cosine theta sine theta, if you want models cos theta, models sine theta. Now, this is still an identity, it's equal.
32:25:50Paolo Guiotto: I want to show now that this is going to zero when XY goes to 0.
32:29:730Paolo Guiotto: So these two costs and signs, they do not help this quantity going to zero.
32:35:270Paolo Guiotto: Because they could be even constant and different from zero.
32:39:160Paolo Guiotto: So they are not of no help. Who is responsible for going to zero is the raw. So, what I do is, I throw away these two guys. I say they are less or equal than 1,
32:50:410Paolo Guiotto: So I can say that this is less or equal than raw. So the distance between the function f at point XY and the candidate limit 0 is less or equal than raw that, remind, is the norm of XY.
33:06:380Paolo Guiotto: And now it is clear that if you send XY to 0, this norm goes to 0,
33:13:580Paolo Guiotto: And by the squeezed argument, the other quantity, which is positive, is squeezed between 0 and someone will push down to zero. So, it must go to zero. So, the distance between F, X, Y
33:29:40Paolo Guiotto: And 0 goes to 0, and this means FXY goes to zero.
33:36:210Paolo Guiotto: So the limit exists, and it is equal to zero.
33:40:360Paolo Guiotto: Okay?
33:42:400Paolo Guiotto: So, what we learned here, is…
33:45:860Paolo Guiotto: Once we use polar coordinates, this, of course, is possible for functions of two variables.
33:54:330Paolo Guiotto: For functions of three variables, we have to understand what are the polar coordinates in space. There are polar coordinates.
34:01:400Paolo Guiotto: They are called the spherical coordinates.
34:04:400Paolo Guiotto: we replace this new coordinate. They emphasize the fact that if with the Cartesian coordinates, going to zero means all the coordinates, they go to zero, but you don't know exactly how.
34:21:230Paolo Guiotto: With polar coordinates, we have still two coordinates. One only is going to zero, the other is just bounded.
34:28:80Paolo Guiotto: So, basically, it's a way to reduce the limit to one variable limit. And when you have done that replacement, you can observe what is F. Here, you have this point here. Well, you can realize that since rho is going to 0, and the other factor is bounded
34:45:20Paolo Guiotto: I must prove that this is going to zero. So I have now the indication that this should be a limit equals zero. So let's see again…
34:56:870Paolo Guiotto: An application of this, for example… But you do number 2… do…
35:09:490Paolo Guiotto: Number two, I do number 3.
35:12:740Paolo Guiotto: which is more or less the same difficulty. We have here limit X, Y, He's going to 0, 0.
35:23:720Paolo Guiotto: X cubed minus Y cubed divided X squared plus Y squared.
35:32:320Paolo Guiotto: Same usual domain, so here… F… is defined…
35:42:600Paolo Guiotto: Domain D, which is, 2.
35:45:680Paolo Guiotto: minus the origin, where we… Wish to compute the limit.
35:52:370Paolo Guiotto: The RNG is therefore an accumulation point, blah blah.
35:56:370Paolo Guiotto: Now… Instead of launching and searching for sections, let's first see what is this thing in polar coordinates.
36:05:390Paolo Guiotto: So, F of XY If I replace X equals raw cos theta.
36:13:600Paolo Guiotto: And Y equal raw sine theta.
36:17:590Paolo Guiotto: Becomes a fraction, so raw cube cosine…
36:24:430Paolo Guiotto: Cosine cube theta minus raw cube sine cubed theta.
36:31:870Paolo Guiotto: Well, of course, you must remind that X squared plus Y squared with this setup is raw square.
36:41:40Paolo Guiotto: And you get this. Now, let's simplify. If possible, you see that we have a common factor raw cube in the denominator. We can't simplify with raw squared, so we get raw times cos cube theta minus sine cube theta.
36:59:280Paolo Guiotto: Okay, I don't know what is this cosine cube, who cares? So this is a number…
37:06:150Paolo Guiotto: which is difference between cosine, sine, cuber. Each of these numbers is between minus 1 and 1, right?
37:14:320Paolo Guiotto: So, I can say that this factor is, at worst, this is 1 and this is minus 1. This cannot be possible. Let's say that if this is 1 and this is minus 1, this parenthesis is equal to plus 2.
37:30:740Paolo Guiotto: at least this is the minimum value to be minus 1 here, and since there is the minus, I take the,
37:39:720Paolo Guiotto: Negative value, most value is minus 1, so it becomes,
37:43:620Paolo Guiotto: Oh, sorry, the highest minus plus 1, so minus 2, okay? So this quantity is…
37:50:750Paolo Guiotto: No larger than 2, not smaller than minus 2. So this means that our FXY, as you can see, is something that goes to 0 times something which is bounded.
38:06:510Paolo Guiotto: And therefore, I would expect that this quantity is small. You remind, when you multiply something that goes to zero with something that, even if it has not any limit, is bounded, the product goes to zero. This is a rule that we have from…
38:19:890Paolo Guiotto: ordinary calculus. So now I bet on the fact that this is, going to zero. So I take an absolute value of FXY,
38:29:140Paolo Guiotto: Minus the limit, that for me should be 0. This is modulus of FXY.
38:35:270Paolo Guiotto: Now, because of this formula, this is raw times models of cos cube theta minus sine cube theta.
38:45:200Paolo Guiotto: At worst, that quantity is plus 2 or minus 2, so an absolute value is no larger than 2.
38:53:300Paolo Guiotto: Okay, so I can say that this is less or equal than 2 raw. That is, returning back to the Cartesian coordinates, 2 times the norm of XY.
39:04:50Paolo Guiotto: And since when XY goes to 0, 0, this quantity goes to 0, when XY Goes to 0, 0.
39:13:510Paolo Guiotto: This will force also this guy to go to zero.
39:17:560Paolo Guiotto: And therefore, this means that if there exists the limit, for XY going to zero.
39:26:420Paolo Guiotto: Zero.
39:28:190Paolo Guiotto: of F, and S is equal to 0.
39:32:300Paolo Guiotto: Okay.
39:33:590Paolo Guiotto: So remind, sections are to disprove existence.
39:38:890Paolo Guiotto: So you cannot obtain… I will repeat, I don't know how many times along the course, but…
39:46:860Paolo Guiotto: 20-30% of you will do this. They will compute the limits along section, they find two sections on which F has the same limit, they conclude that the limit is that value.
39:59:730Paolo Guiotto: Okay? So there will be lots of you that will not understand this point. That's why you should think about a lot on this. So, sections are useful to disprove existence, so I find two sections along which I've asked two different limits. there is no limit.
40:18:690Paolo Guiotto: And that's… and that's it. I cannot use sections to prove existence.
40:23:150Paolo Guiotto: To prove existence, I have to do something like this. So, this is for finite limits, then we will see what happens for infinite limits.
40:32:480Paolo Guiotto: But it's similar, it's not particularly different.
40:36:110Paolo Guiotto: So, you have to evaluate the function and try to reduce to a limit in one variable by using some system of coordinates. That's the unique general strategy.
40:46:90Paolo Guiotto: Now, let's try to draw a general rule. First of all, here we computed the limits at 0, 0.
40:57:370Paolo Guiotto: What if I have a limit in another pointer?
41:01:890Paolo Guiotto: So, let's see what we learned here. So, we learned… That, huh?
41:12:60Paolo Guiotto: to… Ruve.
41:15:300Paolo Guiotto: that limit Well, for XY going to 0, 0,
41:21:790Paolo Guiotto: of FXY equals 0, because in all limits we've got a limit value equals 0. It seems a very particular situation.
41:31:50Paolo Guiotto: So, we have two…
41:38:310Paolo Guiotto: To, obtain, a bound…
41:46:910Paolo Guiotto: So you see that the key point was here. So we evaluate models of F minus 0, which is the candidate limit, that is modulus of F.
41:55:980Paolo Guiotto: So a modulus of FXY Less or equal than what?
42:02:290Paolo Guiotto: Look at the examples. Here, we got that this was less or equal than to the norm of XY.
42:10:70Paolo Guiotto: Previous example, we have got that this was less or equal than norm of XY.
42:18:190Paolo Guiotto: What if it is norm of XY at power 2? So, something like norm of XY squared. It wouldn't change, because when XY goes to 0, norm of XY goes to 0. Also, norm XY power 2 will go to 0.
42:34:620Paolo Guiotto: So what we really need is that this is bounded by a function, let's say G, of the norm of XY,
42:46:330Paolo Guiotto: Such that… G of raw.
42:51:230Paolo Guiotto: goes to zero when… a raw goes to zero.
42:57:890Paolo Guiotto: Because, if G… It's a function of the norm of XY.
43:03:980Paolo Guiotto: that bound from above the absolute value of F, with this property, you are that when XY goes to 0, this quantity, the number of XY, will go to 0. But when the element of G goes to 0, g goes to 0.
43:19:310Paolo Guiotto: So this goes to zero, and therefore, this one will go to zero. But this means that the distance between F and the limiter is going to zero, F is going to the limit.
43:29:360Paolo Guiotto: So now, what if the limit is different from zero? Well, nothing changed, because what we have to estimate is…
43:37:700Paolo Guiotto: To chat, to prove.
43:41:70Paolo Guiotto: that limit of F is equal to L, any number, real, We need… to bound…
43:56:220Paolo Guiotto: So here, we add the limit equals 0. If the limit is non-zero, we have to bound the distance between the function
44:02:820Paolo Guiotto: and the limit.
44:04:550Paolo Guiotto: If we are able to control that distance by a function G of the norm of XY,
44:12:210Paolo Guiotto: With the same property, such that…
44:16:650Paolo Guiotto: G of raw goes to 0.
44:20:30Paolo Guiotto: When rho goes to zero, again, we can conclude, because when XY goes to 0, norm of XY will go to 0, G will go to 0, and therefore, because of this bound, this quantity goes to zero. But the assert band goes to 0 if this goes to this.
44:39:260Paolo Guiotto: And this, provides that.
44:43:770Paolo Guiotto: Now, we can give a vector form to this, because instead of writing XY, let's put the error notation, this means F of
44:54:710Paolo Guiotto: X arrow minus L less than or equal than G of the norm of X arrow.
45:03:440Paolo Guiotto: Wida, again, huh?
45:07:950Paolo Guiotto: with the… G such that G of its argument goes to 0 when the argument row goes to zero.
45:20:400Paolo Guiotto: And this version works, not in R2, works in R3, works in R… in RN, okay, in any number of dimensions. So we have this… I don't do the proof, because it's, it's, let's say, it's more or less intuitive proposition.
45:36:890Paolo Guiotto: So, if we are able to bound f of x minus L.
45:44:610Paolo Guiotto: absolute value. So this is the distance between the value of the function and the value of the limit, okay?
45:52:730Paolo Guiotto: So, distance… Between… F of X.
46:02:370Paolo Guiotto: And, now, if you are able to bound this by a function that we call G,
46:10:780Paolo Guiotto: of the norm of X, huh?
46:14:450Paolo Guiotto: So this is a function, a function.
46:19:480Paolo Guiotto: of norm of X.
46:24:140Paolo Guiotto: Now, this G is not like F. F is a function of an array, G is a function of a numerical quantity, because G of norm of X. Such that…
46:36:510Paolo Guiotto: G of rho goes to 0.
46:40:30Paolo Guiotto: When?
46:42:70Paolo Guiotto: Rho goes to zero.
46:45:50Paolo Guiotto: Then, if this is verified, we have that there exists the limit.
46:52:530Paolo Guiotto: for vector X, Going to vector zero.
46:58:60Paolo Guiotto: of f of x, and this limit is the value L.
47:05:860Paolo Guiotto: This is a test that we can use to,
47:11:20Paolo Guiotto: To compute the limits, at least the limits at zero.
47:17:320Paolo Guiotto: Yeah. It's just an arbitrary function. It's no matter function that goes to zero when the argument goes to zero. So let's see…
47:29:20Paolo Guiotto: Some other example…
47:33:90Paolo Guiotto: Okay, so in exercise 188, there are a couple of examples, which are the last two. They are two-star, means that you must achieve that, but they are not immediate, so let's do one of them, just to understand what happens.
47:49:70Paolo Guiotto: So let's do exercise 1, still 188.
47:55:360Paolo Guiotto: Okay, so since we have done the first 45 minutes, so it's better if we take a break. So, 5 minutes is okay.
48:07:300Paolo Guiotto: Okay, yeah, but let's do 5 minutes, not 15 minutes, because I…
48:13:470Paolo Guiotto: You go around. What happens here?