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00:09:650Paolo Guiotto: Okay, so I left you to finish, or at least to try to finish an examination.
00:15:690Paolo Guiotto: Let's review quickly, then… What is the problem?
00:21:480Paolo Guiotto: And then we… we see other limits. Here, we…
00:31:480Paolo Guiotto: We were computing limits for XY going to 0 of log 1 of 1 plus 2X cubed divided by hyperbolic sine of X squared plus Y squared. We discussed domain, which is a half plane.
00:46:310Paolo Guiotto: Abshesa larger than minus the cubic root of 1 half.
00:51:50Paolo Guiotto: Which is a negative number.
00:53:340Paolo Guiotto: And, minus the .00 where we have to compute the limit.
00:59:150Paolo Guiotto: This figure shows that clearly that point is an accumulation point for the domain, which is colored in yellow, yeah.
01:07:810Paolo Guiotto: Now, we, we started discussing the limit. We noticed that when XY are going to zero, zero means X and Y both going to zero.
01:20:470Paolo Guiotto: And therefore, log is going to log of 1, 0, and hyperbolic sine is going to hyperbolic sine of 0, 0. So, it is a 0 versus 0 in the terminate form.
01:31:790Paolo Guiotto: Feminative form means that there is no general rule to compute the limit.
01:36:330Paolo Guiotto: So we have to, understand how much 0 is the numerator, how much 0 is denominator.
01:43:490Paolo Guiotto: How can we done decent?
01:46:440Paolo Guiotto: Well, for functions of several variables, the usual strategy is to change variables, so use polar coordinates. So here we transformed log 1 plus 2x cubed into this, log 1 plus 2 raw cube cos cube.
02:05:50Paolo Guiotto: This is going to zero because raw is going to zero, so it is like a log of 1 plus something that goes to 0. Here, I reminded of the
02:13:630Paolo Guiotto: McLaurin formula for the logarithm. Log 1 plus t equals t plus little o . That allows us to write that log of 1 plus 2 rho cubed cos cube as 2 rh cubed cos cube plus little O of that quantity.
02:29:280Paolo Guiotto: Now, since it is raw going to zero, and because data is not going anywhere, we may think it's like a constant, doesn't change the size of the little o, we can always eliminate constants into the little o, while it is raw, the quantity going to zero. So, basically, what we have is that, this is log of 1 plus
02:50:960Paolo Guiotto: 2X cubed, it is equal to 2 raw cube cost theta cube plus a little O of rho cubed.
02:58:710Paolo Guiotto: For the denominator, hyperbolic sine x squared plus Y squared, it is sine of rho square after polar coordinates. We're reminded of the Taylor-Maclaurin formula, sine… hyperbolic sine of t equals t plus little O of T, so this becomes rho square plus little O of rho square.
03:16:560Paolo Guiotto: So, this means that our F becomes that fraction.
03:21:140Paolo Guiotto: Since it is raw, the quantity going to zero.
03:25:770Paolo Guiotto: We try to simplify as much as possible. In raw, we can factorize the raw square downstairs, raw cube upstairs, and simplify raw cube with raw squares, so at the end, we have this. Raw times.
03:37:380Paolo Guiotto: pure 2 cos cubed theta, plus a quantity. I wrote epsilon rho, which is the O of raw cube divided raw cube that goes to 0, by definition of little O. And similarly, denominator is 1 plus epsilon rho, where epsilon rho is O of raw square divided raw square that goes to 0, when
03:56:590Paolo Guiotto: Rho goes to zero.
03:57:960Paolo Guiotto: Now, I said we have, at this point, we are betting on the fact that when XY goes to 0, 0 or goes to 0, this quantity will go to 0. So, we proved that F has limit 0. We assess the distance between F and 0. Now, since this means modulus of F and modulus of F is this one.
04:20:990Paolo Guiotto: So when I take the absolute value of all this thing, I get raw, and absolute value of the fraction, which is 2 cos cubed theta plus that quantity that goes to 0, epsilon raw divided 1 plus epsilon raw.
04:38:440Paolo Guiotto: Well, now, that quantity is just bounded, it is not going to zero, so it's not going to help in making that product going to zero, so I just throw away, so, one, raw times the fraction
04:57:520Paolo Guiotto: I put the modulus numerator, modulus denominator. About the numerator, I can say that, this is a modulus of 2 cos cubed, modulus of 2 cos,
05:11:280Paolo Guiotto: cube theta plus this quantity epsilon rho, divide the modulus 1 plus epsilon rho down here.
05:18:310Paolo Guiotto: Now, above, I can use the triangular inequality, say that this is less than 2 mod… modulus 2 cos cubed theta, plus this modulus epsilon rho. Since this is going to zero, we can keep there, and above terminator here.
05:37:500Paolo Guiotto: be careful, because I cannot… I can say that modulus of 1 plus epsilon rho is less or equal than 1, modulus of 1 plus modulus epsilon rho, right.
05:48:830Paolo Guiotto: But, since this is at the luminator.
05:52:720Paolo Guiotto: If I increase the denominator, the fraction decreases while I'm doing a bound, so I need to put something smaller here.
06:02:940Paolo Guiotto: Now, here, there is another inequality, which is still the triangular inequality, that says modulus of A plus beta is, of course, less than modulus of A plus modulus of beta, but it is also larger than modulus of A minus modulus of B.
06:19:580Paolo Guiotto: That's why I need… what I need here, because I can say 1 plus epsilon raw
06:25:580Paolo Guiotto: This is if you want to justify
06:28:410Paolo Guiotto: Well, actually, you don't even need to do that, sorry. We can… we… we… we can just keep 1 plus, because it's constant in… I forgot. Sorry, you can cancel all this.
06:43:490Paolo Guiotto: we don't need to complicate too much stuff, so forget of this, I just write this. So now I can say this is less or equal 1, so this is less or equal rho, 1 plus the absolute value of that epsilon rho divided models 1 plus epsilon rh. The key point is, you know, that these quantities are going to zero.
07:02:780Paolo Guiotto: So, it is clear that this quantity here, when you send the raw to zero, it will go. This one is going to zero.
07:10:160Paolo Guiotto: The epsilon are going to 0, so the fraction is going to 1, so everything is going to 0. So at the end, you can call this is the function G of rh… of the general statement, hmm?
07:25:640Paolo Guiotto: you remind me what I'm talking about. I'm talking about of this,
07:30:950Paolo Guiotto: No, probably we have seen last time.
07:34:480Paolo Guiotto: we have… I'm talking about, of,
07:39:130Paolo Guiotto: this, no? If absolute value of F minus L is controlled by a function of rho that goes to zero when rho goes to zero, we are done, okay? And that's exactly what we have, here.
07:54:460Paolo Guiotto: Okay? So this finishes the exercise.
07:57:970Paolo Guiotto: Okay, I will, since I wrote on the, on the yesterday slide, I will upload this afternoon, the new slide, okay? This will replace the…
08:12:760Paolo Guiotto: the previous one. So, now let's just change the exact… I will do just one more example.
08:22:410Paolo Guiotto: For example, still exercise… 189…
08:33:260Paolo Guiotto: I don't know the number, the number what?
08:38:20Paolo Guiotto: Let's do the number 1.
08:40:750Paolo Guiotto: limit, for XY, still going to… 00 of E…
08:51:140Paolo Guiotto: It's very small here. Wait a second.
08:54:170Paolo Guiotto: Yeah, okay. E to 4Y cubed… minus cos… X squared plus Y squared.
09:05:140Paolo Guiotto: divided by X squared plus Y squared.
09:12:400Paolo Guiotto: So here, domain is simple.
09:15:360Paolo Guiotto: Yeah… F… XY… is defined
09:22:330Paolo Guiotto: As you can see, there is no problem for the exponential, for cosine, for the denominator. The unique problem is the fraction that cannot have a denominator equal zero. So, for…
09:34:330Paolo Guiotto: X, Y…
09:37:60Paolo Guiotto: in domain D, which is the real plane R2, except 00, where we have to compute the limit, by the way.
09:48:440Paolo Guiotto: When XY goes to 0,
09:52:310Paolo Guiotto: we have that both X and Y go to 0. So, in particular, we see that exponential, exponent goes to 0, exponential goes to 1,
10:03:420Paolo Guiotto: Cosine argument goes to 0, cosine goes to 1, so numerator goes to 0.
10:10:360Paolo Guiotto: Okay? Denominator goes to zero, and therefore we have an indeterminate form. We cannot apply any calculus rule, otherwise it would be easy.
10:21:120Paolo Guiotto: Okay, so we have to understand how much is going to zero numerator, and how much is going to zero denominator, and to that, it is convenient to use polar coordinates. So we write F of rh cosine theta, rho sine theta.
10:37:830Paolo Guiotto: These are important coordinates, because one of them is just the norm of the vector. It is raw, which is just going to zero. So we have E24Y cubed, so raw cube sine cubed theta minus cos x squared plus rho square is raw square.
10:58:740Paolo Guiotto: Then we have divided by raw square.
11:03:610Paolo Guiotto: Okay, so denominator, it's explicit, it's just raw square numerator, it's a bit not clear, because we have this combination of,
11:13:930Paolo Guiotto: exponential and cosine. So what can we do? Well, since this exponent is going to 0, we remind that e to t, when t is going to 0, is 1 plus t plus little o .
11:29:400Paolo Guiotto: And similarly, cosine T, when t goes to 0, is 1 minus T squared divided 2 plus O of T squared.
11:38:950Paolo Guiotto: If needed, we can extend these formulas. So we have that denominator.
11:45:100Paolo Guiotto: is equal to, so we applied this formula with t equal 4 raw cube sine cube theta. So this will be 1 plus 4 raw cube sine
12:01:790Paolo Guiotto: cubed theta plus a little o of the same quantity, but since none of 4 and sine are necessarily going to 0, we treat it as a constant, we throw away, and it remains the raw, which is the unique responsible of going to 0.
12:20:750Paolo Guiotto: And, so this is about the exponential E to 4 raw,
12:28:690Paolo Guiotto: cube sine cubed theta. Then we have a minus… what? A minus cos raw square.
12:37:260Paolo Guiotto: plus raw square, which is 1 minus… so here t is raw square for this formula. So rho power 4 divided 2, little o, be careful, because T is raw square, so T squared will be rho power 4.
12:55:650Paolo Guiotto: Okay, so at the end, we have, so we cancel whatever we can. For example, this one goes away with this one. Then we have a 4 raw cube sine…
13:08:230Paolo Guiotto: cubed theta.
13:11:760Paolo Guiotto: plus little o of raw cuba.
13:15:560Paolo Guiotto: Minus, minus, plus raw power 4 divided 2 plus O of rh power 4.
13:22:960Paolo Guiotto: Okay, that's the numerator, it's a bit more clear, and we have to divide by raw square. So F…
13:31:600Paolo Guiotto: is this thing divided by your square. So, numerator…
13:36:300Paolo Guiotto: of a raw square. If we do that, we have 4 raw sine cubed theta plus O of raw cube divided raw square.
13:48:690Paolo Guiotto: was queer.
13:51:250Paolo Guiotto: Plus, rho 4 divided 2 divided by rho square makes rho square divided 2, plus rho power 4 divided rho square.
14:01:220Paolo Guiotto: You know that this little o works like powers. So, for example, this is a little o of…
14:12:940Paolo Guiotto: What is Epsilon?
14:15:360Paolo Guiotto: O of raw cube divided raw square is a little O off.
14:21:980Paolo Guiotto: of Ra. They are like powers. And this is a little of… raw square.
14:30:850Paolo Guiotto: So we have, at the end, 4 rho sine cubed theta plus O of rho plus
14:39:240Paolo Guiotto: Raw square over 2 plus O of raw square.
14:44:130Paolo Guiotto: Here, we may say that since these two are smaller than this one, we can even cancel.
14:50:900Paolo Guiotto: But in any case, now we see that this is going to zero. This is just bounded, so…
14:57:560Paolo Guiotto: it is going to zero. And in fact, distance between F and 0 is equal to this, so modulus for rh sine cubed theta plus little o of rh.
15:13:580Paolo Guiotto: which is, of course, less or equal than. You use the triangular inequality.
15:18:610Paolo Guiotto: And we have models for row sine. So, for rho, modulus sign, which is 1, plus models or rho, which is… we can also throw away the models, since it will be still a quantity going to zero faster than raw.
15:35:440Paolo Guiotto: So this is our G of rho that goes to 0 when rho goes to 0.
15:42:520Paolo Guiotto: And so we did use that. There exists the limit, for XY…
15:50:120Paolo Guiotto: going to 0 of F equals 0. And that's it for this exercise.
15:56:430Paolo Guiotto: So I would say you do… 1… 8… Nice.
16:05:340Paolo Guiotto: Okay, now, so far, so let's say, remark.
16:14:440Paolo Guiotto: So, so far, we computed the…
16:23:760Paolo Guiotto: limits,
16:25:940Paolo Guiotto: let's say, for a vector in two, three variables, so I just write XRO, going to zero of a function f.
16:36:600Paolo Guiotto: equal to a finite limit L.
16:41:610Paolo Guiotto: So now, what if the limit is not in zero?
16:46:50Paolo Guiotto: So, what if the limit is for X going to some point P different from 0? Is that really different? So, first question is.
16:55:570Paolo Guiotto: what… If we have… the limit…
17:03:100Paolo Guiotto: for X going to some point P, F of X, equal L with P.
17:11:990Paolo Guiotto: Different from zero.
17:14:520Paolo Guiotto: Is that so different? Of course, no, because we can always transform any limit at finite point into a limit of 0. We do just a change of variable. We can say that limit when x goes to P,
17:29:240Paolo Guiotto: of F of X, huh?
17:32:990Paolo Guiotto: If X goes to P, you see Y equal X minus P will go to
17:43:690Paolo Guiotto: Zero.
17:44:920Paolo Guiotto: So this will become a limiter
17:48:10Paolo Guiotto: in Y going to zero, of what? Not of F of Y, but of F… what is X in terms of Y? You see, from this relation that X is Y minus P. So you have here Y minus P.
18:06:900Paolo Guiotto: And basically, so you can reduce always a limit to a limiting zero, if it is at some… at a finite point.
18:16:690Paolo Guiotto: At infinity, you cannot do that. It's a different story. We will, we will see in a minute what happens when we have to compute a limit at infinity for a function. By the way, this says that basically it's not a particular restriction
18:35:900Paolo Guiotto: for limited finite points, to consider only the case F equals 0.
18:42:640Paolo Guiotto: Limits at zero are important because an important class of limits are limits at zero, those relative to derivatives.
18:52:60Paolo Guiotto: So, even for one variable calculus, when you compute the derivative, the derivative is limits when H goes to 0, f of x plus h minus f of x divided by H.
19:02:840Paolo Guiotto: That's an important limit in 0. So, limits in zero are important, but not necessarily the unique one. So, the first question is this. Second question is, what if…
19:18:500Paolo Guiotto: What if, we… want… to prove, that…
19:29:750Paolo Guiotto: limit, let's say, on X going to zero of f of x.
19:37:580Paolo Guiotto: Is equal, for example, to plus infinity.
19:43:970Paolo Guiotto: Because, the test we have seen last time, I just reminded a few minutes ago.
19:53:180Paolo Guiotto: The general strategy that we have here works when the limit L is finite.
20:01:410Paolo Guiotto: So, you have your candidate limit L, because you understood that the limit will be that value, finite value.
20:08:690Paolo Guiotto: And, so the strategy is, assess the distance between value f of x and limit.
20:15:400Paolo Guiotto: If you're able to prove that this is controlled by some quantity that depends only on distance to the origin, what we call the raw.
20:23:230Paolo Guiotto: In such a way that when rho goes to zero, that quantity goes to zero, you are done, okay? But of course, you cannot do that with L equal plus infinity minus infinity, things like this, okay? So what can be done in those cases?
20:38:460Paolo Guiotto: Well, if I want to prove that the limit, is going to… for example, the limit is plus infinity, it means F becomes big, no?
20:47:770Paolo Guiotto: So, the, strategy for this case could be the following.
20:56:740Paolo Guiotto: Not to prove that distance between function and limit is small, because this is nonsense.
21:03:240Paolo Guiotto: But prove that function becomes big when you get close to the point, in this case, 0. So I get that by proving something like f of x larger than some G of
21:17:240Paolo Guiotto: norm of X, huh?
21:20:350Paolo Guiotto: with now…
21:30:870Paolo Guiotto: G of rh reminds that since we are doing the limit at 0,
21:37:10Paolo Guiotto: When x goes to 0, the norm of X goes to 0, so raw goes to 0. What should happen to G of rho when rho goes to 0 here?
21:46:590Paolo Guiotto: If I want that, this G pushes up F to plus infinity.
21:52:410Paolo Guiotto: Yes, it must go to Bless infinity, exactly.
21:55:500Paolo Guiotto: Okay? And similarly, If you want to prove
22:02:240Paolo Guiotto: That the limit is minus infinity. So in this case, sorry, let's put a conclusion, otherwise it seems…
22:09:410Paolo Guiotto: Then, in this case, there exists the limit when X goes to zero.
22:16:250Paolo Guiotto: of f of x, and that's equal to plus infinity.
22:20:660Paolo Guiotto: And, similarly.
22:25:810Paolo Guiotto: If we arrive to prove that F of X
22:30:210Paolo Guiotto: is less or equal than any G of norm of X.
22:36:490Paolo Guiotto: We now… this G of rho, going to minus infinity when rho goes to 0,
22:45:750Paolo Guiotto: This will say that F is pushed down by this G.
22:50:560Paolo Guiotto: So, also, F will go to minus infinity, so there exists a limit. When X goes to zero.
22:59:410Paolo Guiotto: of f of x, and that limit is equal to minus infinity.
23:05:790Paolo Guiotto: So, just an example. We don't do so many examples on this type of limits, because it is more interesting when we go to infinity, the infinite limits, but we can do, in any case, an example. Suppose that I want to compute a trivial example.
23:22:750Paolo Guiotto: Limit for XY going to 0, 0 of 1 over X squared plus Y squared.
23:31:240Paolo Guiotto: as you may expect, when both X and Y go to 0, this quantity should go to plus infinity, because terminator goes to 0. So, this is my F,
23:41:550Paolo Guiotto: XY, you notice that the strategy is the same, so you use polar coordinates because they emphasize the variable which is going to zero, really, which is raw. So, raw cosine theta, raw sine theta.
24:00:890Paolo Guiotto: You put inside this, and this comes exactly a function of one variable, because it is one of a rho square. That's your G of rho.
24:11:70Paolo Guiotto: that does exactly what we want, so this goes to plus infinity when rho goes to zero. Of course, we say rho goes to zero, but remind, rho is a positive variable, and not be negative, so rho zero plus.
24:28:10Paolo Guiotto: And so we concluded that, from this, there exists a limit when XY
24:34:250Paolo Guiotto: goes to vector 0 of F equal to plus infinity.
24:43:610Paolo Guiotto: So, as you can see from these first examples, what is going on is that we do not have a really direct method you compute, as in the first year calculus, you compute the limit by doing a certain number of algebraic steps.
25:01:620Paolo Guiotto: But you have to… to try to obtain a bound of the function from below, from above, it depends on what you want to prove, with something that depends on the distance to the point.
25:19:120Paolo Guiotto: Okay? This is the strategy.
25:22:280Paolo Guiotto: Now, let's come to infinite limits. Sorry, limits at infinite.
25:29:490Paolo Guiotto: So, limits… at… The infinity of the space.
25:37:390Paolo Guiotto: Now, let's go just a second back to the definition of limit to realize that this case is included in the definition.
25:48:00Paolo Guiotto: Because, if we go back to the definition of limit for a function.
25:53:720Paolo Guiotto: Which is not… yeah, more or less sincere, but let's go back to this one.
25:59:290Paolo Guiotto: Okay, so we said that the limit when X goes to point P is L. This can be even infinite, the value, but we're not talking about the value of the limit, but the point
26:12:610Paolo Guiotto: Infinity, okay? Now, we are taking limits when x goes to infinity.
26:17:920Paolo Guiotto: Well, the definition works also for the infinity, because it says whenever you take a sequence in the domain, different from the infinity, this
26:26:600Paolo Guiotto: will be a redundant condition, because points are in the domain, they are in the space, they are not infinity, so this is automatic for the infinity, okay? But all the remaining remains… so, for every sequence in domain that goes to infinity, f of xn goes to the limit n.
26:45:550Paolo Guiotto: So the sequence… the definition works… contains already that case.
26:53:120Paolo Guiotto: And this means that the same kind of arguments will apply also to these limits. So let's start…
27:02:390Paolo Guiotto: With some example.
27:06:260Paolo Guiotto: Okay, let's take… this one.
27:15:130Paolo Guiotto: Let's take one… someone which is not particularly… okay.
27:20:470Paolo Guiotto: This is the example 1 for 12.
27:26:360Paolo Guiotto: So, the question is, show that, the limit… when XY…
27:37:50Paolo Guiotto: goes to the infinity of R2.
27:40:140Paolo Guiotto: of this, X squared plus Y squared minus 4XY, does not exist.
27:50:220Paolo Guiotto: Now, the argument is always the same. To prove that a limit does not exist, I just need to find two ways to go to the limit point, which is in this case, infinity.
28:01:740Paolo Guiotto: Along which, the function has two different limits.
28:05:960Paolo Guiotto: Okay? So the unique difference is that, until now, we considered only 0.00, so all roads were going to 00. Now, since we are going to the infinity, the roads must go to infinity. Okay? So this means far away.
28:22:660Paolo Guiotto: Let's see what can be done here. So, here, the function f is, of course, that function which has no problem.
28:32:420Paolo Guiotto: So, it's X squared plus Y squared minus 4XY.
28:40:50Paolo Guiotto: F is defined… on domain D, which is the full plane R2, so no problem.
28:50:360Paolo Guiotto: And it is clear that infinity is an accumulation point for this, because there exists a sequence of point over 2 that goes to infinity. There are… it's plenty of sequences. Okay, you cannot visualize the infinity, because the infinity is somewhere far away.
29:06:400Paolo Guiotto: But it's everywhere far away, so it's not a point. You cannot see where is it, this infinity. But this means that if you take, for example, take the sequence of points and zero, this sequence is going to infinity in the plane. So this is a sequence of points of the domain that goes to infinity.
29:24:820Paolo Guiotto: So this means that infinity is an accumulation point for this domain D, and the definition of limit makes sense.
29:36:300Paolo Guiotto: Okay, so now, we take, as exactly as in the case of limits at finite point, we take restrictions, we look at restrictions of this F,
29:49:60Paolo Guiotto: to see what happens. For example, we could use the axis, as in the case of limited 0. The unique difference is that when now you have a point X0, if you want to send this point to infinity in the plane.
30:04:600Paolo Guiotto: This means that, norm of X0, Go to plus infinity.
30:13:740Paolo Guiotto: A norm of X0 is…
30:20:730Paolo Guiotto: Ease?
30:22:850Paolo Guiotto: is not X.
30:24:980Paolo Guiotto: modulus of X.
30:26:860Paolo Guiotto: Okay, this means the modulus of X going to plus infinity. So, X can go to plus infinity, so in that case, you will move that way, but also you could go to minus infinity. In that case, you will move that way.
30:41:680Paolo Guiotto: In any case, it will go to infinity in the plane, or infinity in the plane. That is not the plus infinity and minus infinity here.
30:48:610Paolo Guiotto: That is a unique infinity.
30:50:850Paolo Guiotto: Okay, so let's see what happens when we evaluate F along these points.
30:56:410Paolo Guiotto: FX0. F is very easy, X squared plus Y squared, so you get immediately X squared.
31:03:760Paolo Guiotto: No? Because Y is 0, so you see that Y squared and 4XY are both equal to 0.
31:10:990Paolo Guiotto: And since we want to send this point X0, to infinity, we send the absolute value of X to plus infinity, so what will happen to X squared when modulus of X goes to plus infinity?
31:29:550Paolo Guiotto: It goes to plus infinity.
31:32:120Paolo Guiotto: Now, as you can see, this function is perfectly symmetric in X and Y. If you flip X with Y, nothing changes. So, it's completely useless to look at section 0Y, because it will provide the same behavior.
31:46:190Paolo Guiotto: So if we want to find, since the scope of the exercise is to show that the limit does not exist, we need to find another way of going at infinity, along which we have a limit which is different from this one.
32:00:790Paolo Guiotto: So we… another way that goes to plus infinity is not interesting.
32:06:90Paolo Guiotto: So what could be another way to go at infinity along which the function does not go to plus infinity?
32:13:760Paolo Guiotto: Do you see anyone?
32:15:450Paolo Guiotto: Excellent.
32:16:550Paolo Guiotto: Exactly.
32:17:900Paolo Guiotto: Because if you evaluate F at XX, this means that we are now running along this straight line.
32:27:510Paolo Guiotto: So we are here, or maybe here.
32:30:530Paolo Guiotto: X can be positive, negative. In any case, point XX goes to infinity in plane. If and all if, the condition is always the same. The norm of the point goes to plus infinity.
32:46:130Paolo Guiotto: But when this happens.
32:48:630Paolo Guiotto: If you compute the norm, this is root of X squared plus x squared, so it is root of 2x squared, which is root of 2 times modulus of X.
33:00:990Paolo Guiotto: So at the end, apart for the root of 2, which is a constant, it is modulus of X that must go to plus infinity, so it's the same thing.
33:10:700Paolo Guiotto: This means that if X goes to plus infinity, the points move upward in the first quarter. If X goes down to negative infinity, the points move down to the left, no? But in any case, you are going far away from the origin, that's the point.
33:26:240Paolo Guiotto: Okay, now when we evaluate this, we get X squared plus Y squared, which is again X squared, minus 4X times Y, which is X.
33:34:880Paolo Guiotto: So at the end, we have 2X squared minus 4X squared. This is minus 2X squared.
33:42:440Paolo Guiotto: And so, when absolute value of X goes to plus infinity, this quantity goes to minus infinity.
33:49:210Paolo Guiotto: So this is the value which is different from… What is it? This one?
33:57:290Paolo Guiotto: So we concluded that this function has no limits.
34:09:639Paolo Guiotto: At infinity, of course.
34:12:480Paolo Guiotto: Okay?
34:15:730Paolo Guiotto: Okay, let me see if there is a…
34:23:730Paolo Guiotto: So, the exercise… no, the exerciser is the 1810,
34:29:900Paolo Guiotto: We will do later some of this. In this exercise, you have to discuss.
34:38:290Paolo Guiotto: If it exists or not, if it exists, you have to compute the value. Let me pick one for which we do not have a limit. Number one.
34:49:310Paolo Guiotto: This is the limit.
34:51:580Paolo Guiotto: for XY going to infinity in,
34:56:60Paolo Guiotto: plane of X cubed plus XY squared minus Y squared.
35:03:880Paolo Guiotto: So, of course, we start with simple functions, like polynomials, then we can extend to any kind of function.
35:11:130Paolo Guiotto: So, also, here you see that the function f is defined
35:17:870Paolo Guiotto: on domain D, which is the full plane R2, again.
35:24:720Paolo Guiotto: So, infinity is an accumulation point, blah blah blah.
35:28:700Paolo Guiotto: Now, if we look at sections, it is clear that, as I said, natural sections are those along the axis, because one or more coordinates are zero, so maybe this simplifies something.
35:42:20Paolo Guiotto: But that's not a… it's not for granted that we get, we get something. FX0, in this case, is X cubed.
35:51:550Paolo Guiotto: And that's interesting because, look, point X0, we already have seen this, but let's remind, X0 goes to infinity in plane if and if…
36:07:170Paolo Guiotto: Not X. Models of X.
36:10:560Paolo Guiotto: goes to plus infinity. So this yields two possibilities, basically. Either X goes to plus infinity, or X goes to minus infinity.
36:22:410Paolo Guiotto: So actually, this can be seen as two ways to go to infinity. One is going along the positive direction of the x-axis.
36:32:330Paolo Guiotto: The other is along the negative direction. These are two ways.
36:37:590Paolo Guiotto: And in fact, here we have a difference, because if we go along the red way, so X is going to plus infinity, X cubed is going to plus infinity.
36:48:50Paolo Guiotto: So when X is going to plus infinity. So if you move the point X0 to infinity in plane, and you evaluate F, the function will lead you to plus infinity.
36:59:770Paolo Guiotto: But, if you move the point along the green line, so here.
37:06:370Paolo Guiotto: So moving to the left, in this case, so this means that X is going to negative infinity. The limit of X cubed will be minus infinity. So as you can see, they are different.
37:20:330Paolo Guiotto: And so this says that there is no limit.
37:24:580Paolo Guiotto: at infinity for this F.
37:30:200Paolo Guiotto: Okay?
37:33:880Paolo Guiotto: Okay, so basically, as you can see, the strategy is the same.
37:39:990Paolo Guiotto: You have to, look for sections.
37:44:130Paolo Guiotto: Two sections, at least, along which you have two different limits.
37:49:660Paolo Guiotto: How can I find these sections? There is not a rule.
37:53:810Paolo Guiotto: Okay, it means you can be lucky, the problem is easy, simple sections work, but the problem maybe is complex, and you have to fight a bit to understand what is the right thing to do.
38:12:30Paolo Guiotto: Now, let's see, how to do…
38:15:510Paolo Guiotto: To prove that a limit exists.
38:19:480Paolo Guiotto: Not always, of course, but often, I mean, for example, when we deal with polynomials, possible limits can be infinite, because these variables are going to infinity.
38:33:220Paolo Guiotto: So this is the kind of examples we are now going to consider. Let me start with an example, which is… I don't think is in the notes. So, let's suppose that we look at this limit…
38:45:390Paolo Guiotto: for XY going to infinity in plane of X squared plus Y squared minus XY.
39:01:330Paolo Guiotto: So again, here, our F, which is this, F is well defined.
39:09:670Paolo Guiotto: on D equal to R2, Infinity is an accumulation point, so the problem makes sense.
39:18:580Paolo Guiotto: If you look at sections, so simple section says that FX0 is equal to X squared, it goes to plus infinity.
39:27:860Paolo Guiotto: when modulus of X goes to plus infinity, remind that modulus of X for this case means point X0 goes to the infinity in play.
39:40:990Paolo Guiotto: If I look at the Y section, because of the perfect symmetry of the function, nothing changed. If I look on straight sections, so something like XMX, so I'm now moving along a line like that.
39:56:530Paolo Guiotto: Y equal MX. My point will be X comma MX. And of course, what do you expect? That point X
40:08:420Paolo Guiotto: Amex.
40:09:890Paolo Guiotto: goes to infinity.
40:12:940Paolo Guiotto: If and only if… So I need to compute, I hope, what should happen to go to infinity here?
40:23:390Paolo Guiotto: Yeah, either X goes to plus infinity, or X goes to minus infinity. In any case, models of X goes to plus infinity.
40:31:480Paolo Guiotto: Now, if we do the calculation here, we have X squared plus
40:35:860Paolo Guiotto: M squared X squared minus X times Y, so I would say MX squared.
40:43:370Paolo Guiotto: So all these,
40:47:470Paolo Guiotto: is, you see an X square multiplied by M squared minus M plus 1.
40:57:680Paolo Guiotto: Okay, so you see that that M is freezed here, is a coefficient, fixed, while what is variable is this one. Definitely this goes to plus infinity. But what about this? This is a constant.
41:12:850Paolo Guiotto: So, if you want to find something interesting, you should have that this quantity can be positive, can be negative. So, in that case, I would say, if for some M, this is positive, for those M, limit is plus infinity.
41:27:420Paolo Guiotto: If for some other M, this quantity is negative, that limit will be minus infinity. So this means that there could be different M along which I have different limits. This could be used to disprove existence.
41:40:990Paolo Guiotto: But what can we say about this quantity?
41:43:660Paolo Guiotto: Well, if you look, this is a second-degree polynomial.
41:46:950Paolo Guiotto: And, if you compute the delta, delta is minus 1 square minus 4 times 1 times 1, it is minus 3, it is negative.
41:58:590Paolo Guiotto: So this parabola never crosses the axis, and since the second degree coefficient is 1, it means that the parabola is always above the axis. So this is saying that M squared minus M plus 1, whatever is M is positive.
42:17:150Paolo Guiotto: So this means that this quantity, F of X,
42:22:790Paolo Guiotto: X will not yield any surprise. It will go to plus infinity.
42:29:610Paolo Guiotto: Now, as you have seen with the limits we have done at zero, even if we know that along every straight line, things go nicely.
42:40:280Paolo Guiotto: We cannot conclude that the limit is that value, okay?
42:47:580Paolo Guiotto: So, now let's go back, understand if it is worth to waste time looking for strange sections along which this function has a different limit than plus infinity, or it makes sense to try to prove that this has limit
43:06:500Paolo Guiotto: Equal, we know what should be the limit.
43:09:890Paolo Guiotto: Because this is the unique possibility, no?
43:15:30Paolo Guiotto: Now, what can be said here? Clearly, since point XY is going to infinity, remind that going to infinity is different than going to any other point.
43:26:450Paolo Guiotto: Because going to any point, finite point, means that each of the coordinates must go to the coordinate of the limit.
43:35:600Paolo Guiotto: Going at infinity could mean that none of the coordinates has a limit, no? You remind the point that goes far away from the origin, moving along a spiral, so none of the coordinates has a limit, but the point goes to infinity. So we have to be a little bit more careful, because this does not mean that X is going to infinity and Y is going to infinity.
43:59:230Paolo Guiotto: So I cannot say, this is going to infinity, this is going to infinity, this is wrong.
44:03:950Paolo Guiotto: What can I say is that this quantity, however, X squared plus Y squared is…
44:12:490Paolo Guiotto: Yes, what is it? X squared plus Y squared is… These…
44:20:190Paolo Guiotto: with respect to XY, it is not the norm, but it is the square of the norm, no? So this is the raw square, if you want.
44:28:540Paolo Guiotto: And the norm is the quantity going to infinity. So, this thing means norm of XY goes to plus infinity. So, in particular, this means X squared plus Y squared goes to infinity. So this guy is going definitely to infinity. While this might not
44:47:470Paolo Guiotto: Imagine you move along the axis, one of the two coordinates is 0, for example, x0. Product XY is constantly equal to 0. So X times Y is not going to infinity.
45:00:250Paolo Guiotto: Okay? But it may go to infinity. For example, x equals y, this is a quadratic term, so maybe, since there is a minus, it's going to bother the other one. So how can we say this? Well, the solution is always the same. Let's look at this function in polar coordinate, and let's see what is the…
45:19:210Paolo Guiotto: So let's finish this, then we take it back.
45:22:60Paolo Guiotto: So, this means we use for X raw cosine theta, for Y raw sine theta.
45:29:580Paolo Guiotto: Remind that,
45:32:50Paolo Guiotto: So, point XY goes to infinity in plain, if and only if this means norm of XY goes to plus infinity.
45:44:650Paolo Guiotto: This means that root of X squared plus Y squared goes to plus infinity.
45:52:520Paolo Guiotto: And that's rooted in the geometry of polar coordinates.
45:57:830Paolo Guiotto: If this is point XY, this is raw, this is theta. This means exactly raw goes to plus infinity.
46:06:510Paolo Guiotto: What about theta?
46:08:390Paolo Guiotto: Nothing else than theta is just in the interval 0, 2 pi.
46:14:380Paolo Guiotto: So, what is the difference with going to zero? That rho goes to zero? Here, rho goes to plus infinity.
46:20:530Paolo Guiotto: Okay?
46:21:940Paolo Guiotto: So, when we do that, we see that the quantity X squared plus Y squared is raw square, so this becomes raw square, minus X times Y, which is raw square times sine theta
46:37:190Paolo Guiotto: Cos theta.
46:39:240Paolo Guiotto: So, we have…
46:45:850Paolo Guiotto: that our F is equal to rho square, I factorized this raw square, times 1 minus sine theta
46:56:730Paolo Guiotto: Cosine theta.
46:59:690Paolo Guiotto: Okay, let's look at this. This is F is equal to, is not an approximation. It's not a bound, it's an identity. This is F written with other coordinates.
47:10:600Paolo Guiotto: Maybe it's worse, maybe it's better. It depends on what we have to do. For limits, it's better, because here I see that this guy is going to infinity.
47:20:630Paolo Guiotto: What about this parenthesis? When I multiply, someone who is going to infinity by someone else
47:27:700Paolo Guiotto: What I need to be aware is that if someone else is 0, this is a problem. You see? Because if this factor is 0, I destroy this rho square that goes to infinity, and I have an indeterminate form.
47:44:730Paolo Guiotto: So, can this quantity be zero, for example?
47:48:570Paolo Guiotto: Well, to have this parentheses equal 0, we need sine times cos equals 1.
47:54:370Paolo Guiotto: Now, sine and cosine are both between minus 1 and 1.
47:59:150Paolo Guiotto: So when you have 1? When both are equal to 1, or both are equal to minus 1?
48:05:410Paolo Guiotto: So the question is, is it possible that both sine and cosine are equal to 1?
48:11:20Paolo Guiotto: No, because sine is 1 at pi half, for example, where cosine is 0.
48:17:70Paolo Guiotto: And cosine is 1 at 0, where sine is 0. So you see that that seems to be impossible.
48:22:870Paolo Guiotto: Okay, but what if this quantity is non-zero, but gets close to zero? That would be still a danger, because if it is close to zero, I still will destroy this throw square.
48:35:660Paolo Guiotto: Now, I need to do something here. Well, this something can be done in this case because of what you know about sine and cosine, and this is the duplication formula. You can write 1 half and 2 here in such a way that this becomes sine of 2 theta.
48:52:280Paolo Guiotto: If I do this, I have that my F becomes rho squared times 1 minus half of sine 2 theta.
49:01:660Paolo Guiotto: What's changed here? A lot.
49:04:230Paolo Guiotto: Because now you see that this term here, which is the same of this one.
49:08:850Paolo Guiotto: Here you see much better that this does not bother too much.
49:13:420Paolo Guiotto: Because, this one-half sign, huh?
49:16:740Paolo Guiotto: is obstulating between, at worst, minus 1 half, at most plus one half, because of this factor. So you're taking 1 minus something, which is between minus 1 half and plus 1 half. So, this 1 minus will be between
49:36:10Paolo Guiotto: The smallest value is when this is plus 1 half, so you subtract 1 half.
49:41:700Paolo Guiotto: So it comes 1 minus 1 half is 1 half. So this parenthesis is sadly greater than 1 half, and no larger than
49:54:510Paolo Guiotto: What the biggest value you can get here, because of this minus, is when sine is minus.
50:01:400Paolo Guiotto: Not when sine is 1, no? When sine is minus 1, this becomes a plus, so 1 plus 1 half, so at most, you get 3 halves.
50:11:630Paolo Guiotto: So you have that parenthesis is varying, maybe, with theta, but it's not going to be close to zero.
50:20:490Paolo Guiotto: And actually, this is irrelevant here, but we can say that it is not even too big. It's bounded above by 3F. So this says that our F…
50:35:200Paolo Guiotto: Is, at most, at least.
50:38:220Paolo Guiotto: At most, I take the bigger bound, 3 halves, so it is no bigger than 3 halves raw square. And at least I take the smallest bound, so it is at least 1 half raw square.
50:53:400Paolo Guiotto: Now, what does it mean, this?
50:55:640Paolo Guiotto: Since, remind, XY goes to infinity, if and only if raw is going to plus infinity.
51:07:90Paolo Guiotto: And you see that. When XY goes to infinity, raw goes to plus infinity, so the two guys here both go to plus infinity.
51:15:840Paolo Guiotto: So they are the two cups that push F to plus infinity. So it's again the squeeze theorem, basically. Then it follows that this guy goes to plus infinity, this one goes to plus infinity, and therefore we deduce that there exists the limit
51:33:710Paolo Guiotto: for XY, Going to infinity.
51:38:430Paolo Guiotto: of F, and this is equal to plus infinity.
51:43:940Paolo Guiotto: Okay.
51:46:460Paolo Guiotto: Okay, so, let's take a short break.
51:51:990Paolo Guiotto: And then we continue. Do you want the 5 minutes, 10 minutes?
51:58:520Paolo Guiotto: 10… Okay, let's do 10 minutes.
53:09:340Paolo Guiotto: Turned after the change of variable is still a function of two variables.
53:14:450Paolo Guiotto: So, the point is that to arrive to a function of one variable, which is the raw, you must eliminate, somehow, the other variables. Here, there is just one variable, the theta, and how do you eliminate? With bounds.
53:30:410Paolo Guiotto: Which is exactly what we did here. So you see that at the end, you don't see anymore the theta.
53:36:520Paolo Guiotto: There is only raw, which is going to infinity, so this, in this case, obtaining this bound says that you are bounded by two objects that are both pushing to plus infinity, you will go to plus infinity. This is the mechanism of the squeeze theorem.
53:56:40Paolo Guiotto: Okay?
53:58:60Paolo Guiotto: So, let's see this in action once more. For example, on a limiting tree variables.
54:06:30Paolo Guiotto: Example 1.
54:08:350Paolo Guiotto: 4… 14, huh?
54:11:590Paolo Guiotto: We have a limit.
54:15:200Paolo Guiotto: or XYZ, Going to infinity 3.
54:21:690Paolo Guiotto: Off.
54:22:730Paolo Guiotto: X squared plus Y squared plus Z squared, squared minus XYZ.
54:32:440Paolo Guiotto: Well, these examples are relatively easy, okay?
54:37:690Paolo Guiotto: So the point is, do not focus on the specific calculations, but focus on the idea.
54:44:740Paolo Guiotto: Then, when we add the technical complexity, you can enter into another kind of level, but the idea will be always the same.
54:53:870Paolo Guiotto: Okay?
54:55:130Paolo Guiotto: So, here, this is our F.
54:59:880Paolo Guiotto: XYZ… Of course, it is defined everywhere as… is defined… on… R3, This is our domain.
55:13:840Paolo Guiotto: Of course, the infinity of a tree is an accumulation point for this domain.
55:19:280Paolo Guiotto: If you want to look, to a section, as you can see, this function is perfectly symmetric in XYZ, no? So what happens for Z… for X will happen for Y and Z.
55:33:170Paolo Guiotto: So this is X to power 4 minus 0, so it will go to plus infinity when point X00 goes to the infinity in R3, and this means, of course, modulus of X goes to plus infinity.
55:52:440Paolo Guiotto: So this, at this point, says that if there is a limit, this limit will be plus infinity. There is no other possibility, okay? This because, remind, if there is a limit along all sections, so all the way going to infinity, you must go to the same value.
56:09:820Paolo Guiotto: So if there is a value, that value must be plus infinity. Now.
56:14:160Paolo Guiotto: Where can we bet? I told you it is better always try to start unless you see immediately that something can go wrong, and so you see immediately some sections, some bad sections. Otherwise, if you don't have any idea, let's see what happens with the
56:32:670Paolo Guiotto: system of spatial coordinates, where one of them is the norm, okay?
56:38:500Paolo Guiotto: And so here we use those called spherical coordinates. So in, spherical… coordinates.
56:51:330Paolo Guiotto: So we introduced the last time. I remind you that we identified the position of point XYZ.
57:01:450Paolo Guiotto: through the distance to the origin, an angle phi, and with the z-axis, and an angle made by the orthogonal projection on the remaining plane, XY, and the X axis.
57:16:40Paolo Guiotto: There are cylindrical… sorry, spherical coordinates where you, for example, could use, because maybe you have a convenience in that, you have your point XYZ, and instead of using the z-axis, you can use the Y axis. So it means that there will be an angle phi between the vector
57:40:280Paolo Guiotto: and the y-axis, and then you will project on the orthogonal plane. So, in this case, on the plane XZ.
57:47:700Paolo Guiotto: So you will have something like D here, and maybe you use a… well, if you use the positive direction in this figure, the angle should…
57:57:760Paolo Guiotto: Should be something like this one.
58:01:120Paolo Guiotto: Okay, so this is the angle theta. But in any case, it's the same philosophy, okay? So let's use this one. This means that X, Y, and Z are. So since this is based on the Z axis, this is Rokos phi
58:16:910Paolo Guiotto: These are raw sine phi.
58:20:20Paolo Guiotto: Raw sine phi.
58:22:550Paolo Guiotto: And then we have cos theta sine theta.
58:26:120Paolo Guiotto: Just to make clear, for the other example, since here we are basing on the y-axis, it is the Y to be Rokos Phi.
58:36:670Paolo Guiotto: The other two, X and Z, are raw sine phi, Raw sine phi…
58:44:500Paolo Guiotto: And this will be cos theta at this sine theta. As you can see.
58:49:180Paolo Guiotto: if you change the three letters, X, YZ, into ZYX, you get the same, no, so it's just a formal adjustment. So we will normally use this, even if there is an asymmetry, no, between the three variables.
59:09:290Paolo Guiotto: Okay, so, by using this one, F.
59:15:470Paolo Guiotto: The important thing to know is that raw.
59:18:540Paolo Guiotto: is, of course, the norm of XYZ.
59:22:630Paolo Guiotto: In any case, even for that one, so it will be always the square root of X squared plus Y squared plus Z square.
59:31:440Paolo Guiotto: So when this system of coordinates is interesting when functions depends on that quantity, on the distance to the origin, because this simplifies a lot, no? So there is a symmetry with respect to the origin. In fact, our function will be raw square, which is that X squared plus Y squared plus Z squared.
59:55:440Paolo Guiotto: Then I have to square, again.
59:59:210Paolo Guiotto: Minus XYZ. Unfortunately, I have to write this. So, since XYZ are all multiplied by raw, I get a raw cube.
00:09:780Paolo Guiotto: Then I have sine phi cos theta from X. Then, from Y, I have another sine phi, so sine phi squared, sine theta.
00:20:510Paolo Guiotto: Times theta, and from that, I have a cost fee.
00:25:540Paolo Guiotto: I have all these things.
00:27:590Paolo Guiotto: So, at the end, what I see is rho power 4, so let's emphasize these things, rho power 4 minus
00:36:600Paolo Guiotto: Rho power 3, and then we have a bunch of factors, sine square phi, cos theta, sine
00:47:400Paolo Guiotto: theta cos phi. Now, if you look at this formula, this formula, you have to look how it depends on draw.
00:57:370Paolo Guiotto: So it's not a polynomial. It's not rho to power 4 minus rho to power 3.
01:02:790Paolo Guiotto: Okay? It's not like that. If this… if this would be our F, we would see immediately, well, it is rho power 4 dominating, because rho now is going to infinity.
01:15:510Paolo Guiotto: Remember that when you go to infinity, powers are bigger when the exponent is bigger.
01:20:460Paolo Guiotto: When you go to zero, power has smaller, and the exponent is bigger. Okay? So…
01:27:650Paolo Guiotto: Here we have this, long coefficient that, however, even if it depends on the other variables, theta and the phi, is something which is a bounded stuff.
01:40:960Paolo Guiotto: Because it's product of many factors, all of them vary between minus 1 and 1.
01:46:270Paolo Guiotto: So, at most, I don't want to be… maybe it's impossible that I get one here, because to get 1, everything must be equal to 1, since they are less than 1. And, for example, you see that cosine and sine cannot be both equal 1, or they cannot be both equal minus 1, or they cannot be 1 and minus 1, so these values are not allowed here, but…
02:10:620Paolo Guiotto: I don't need to be such precise. I just need here. That's because this is the factor of raw cube, and in any case, raw power 4 will kill this.
02:21:820Paolo Guiotto: going to infinity. So here, I need just a rough bound, which is… all this box, Here.
02:32:110Paolo Guiotto: is, at most, Plus 1, at least minus 1.
02:37:720Paolo Guiotto: So, I don't know what is that factor, but I'm multiplying by a quantity which is between minus 1 and 1, so this means that my F will be, at most, at least.
02:49:110Paolo Guiotto: The worst case is when the box is minus 1, because of the minus in front of row cube. So this becomes… the biggest value is rho power 4 plus rho cube.
03:01:60Paolo Guiotto: The smallest value is when that box is plus 1, so I have raw power 4 minus raw cube.
03:07:790Paolo Guiotto: You see? Now, I have bounded my F, which is a function of three variables, with a function which depends, yes, on three variables, because do not forget that rho is the root of X squared plus Y squared plus Z squared.
03:21:690Paolo Guiotto: But it's in fact a function of one variable, raw, that goes to infinity plus infinity when raw goes to plus infinity.
03:33:640Paolo Guiotto: Actually, you may notice that this bound here is useless.
03:41:130Paolo Guiotto: That's the important bound.
03:46:500Paolo Guiotto: Because, you see, in this story, even if I don't know that there is this one, from this side, I know that F is pushed from below by someone that goes up to plus infinity. So it must go to plus infinity.
04:02:40Paolo Guiotto: So, actually, you just need to prove this one.
04:05:860Paolo Guiotto: And that's the difficult part, in fact, of the story, normally. Because that one, as you can see, it's an upper bound, so you know that everything is less than, and that's enough, more or less.
04:18:290Paolo Guiotto: The lower bound is always a little bit more complex, sometimes can be much more complicated. We have an example here, no? Look at the previous exercise. When we computed F, we had this.
04:32:250Paolo Guiotto: Well, we can say this is no more than all this, this is one, let's say that this is minus 1, it was no more than two, but two Ks. The difficulty is to show that this is bounded away from zero.
04:45:240Paolo Guiotto: Otherwise, I can risk to destroy this thing. So the difficulty is to prove that this is actually greater than something positive.
04:54:550Paolo Guiotto: And to do that, we had to do a little trick, which is this one, right? This has 1 half sine of 2 theta, so something slightly non-trivial. Of course, we are talking about something that everyone knows, no? The duplication formula of sine.
05:11:220Paolo Guiotto: It's not something which is extremely difficult. But you see, there is a non-trivial step here, no?
05:19:450Paolo Guiotto: In this case, it's much easier. Why? Because we have this one.
05:25:150Paolo Guiotto: Which is alone dominating this.
05:27:790Paolo Guiotto: So we don't have to be too much refined to get that bound. Okay, so from this, we can obtain that there exists a limit
05:39:460Paolo Guiotto: when X, Y, Z goes to infinity of the space of F and is equal to plus infinity.
05:48:320Paolo Guiotto: So, let's see what is the lesson we can learn here. So, the general factor…
05:54:110Paolo Guiotto: proposition, we do not prove this, but let's, let's, let's write this as a statement. So if we know that F of vector X, so let's try to write in a general form.
06:09:520Paolo Guiotto: is bounded, that's what we need. From below, by a function g of norm of X,
06:18:860Paolo Guiotto: such that this G of rho goes to plus infinity when rho goes to plus infinity.
06:29:120Paolo Guiotto: Okay, so the G of raw is this guy here.
06:35:550Paolo Guiotto: It's not this part.
06:38:40Paolo Guiotto: Okay?
06:39:330Paolo Guiotto: Then, we can conclude that there exists the limit
06:43:750Paolo Guiotto: for X going to infinity of…
06:47:770Paolo Guiotto: the space of F of X,
06:51:200Paolo Guiotto: And that limit is equal to plus infinity.
06:54:250Paolo Guiotto: Similarly, What will be the version of this for limit minus infinity?
07:02:960Paolo Guiotto: If the function f It's now bonded from above.
07:09:90Paolo Guiotto: by a G of the norm of X.
07:12:730Paolo Guiotto: Such that I want to go to minus infinity. Now, my G, the upper bound, goes to minus infinity when rho goes to plus infinity.
07:24:790Paolo Guiotto: Then, in this case, I will conclude that there exists a limit for x going to infinity of f of x, and that limit is minus infinity.
07:38:140Paolo Guiotto: Okay, so this is the general rule that can be proved. It's not particularly complicated proof. If you want, you can try to do that as an exercise.
07:50:100Paolo Guiotto: By putting together the elements, so the definition of limit, what does it mean that the limit is plus infinity, and so on.
07:57:810Paolo Guiotto: Okay? Maybe it could be nice… dry… 2… Right.
08:07:640Paolo Guiotto: a proof, and I… if I…
08:11:350Paolo Guiotto: If I remind this, I will write the proof in the solutions, okay?
08:15:840Paolo Guiotto: So try to write the proof. It's not a particularly important proof, but it's just to see if you are able to master these topics.
08:28:359Paolo Guiotto: Now, I said that, There is an exercise here.
08:35:670Paolo Guiotto: Which is the 1810.
08:40:950Paolo Guiotto: Okay, you can do all of them. Let me show another example here.
08:47:640Paolo Guiotto: Which is, now, to see something which is a little bit more spicy.
08:53:710Paolo Guiotto: You know, you come from Middle East, you like spices still. So, let's see what is spicy here.
09:00:830Paolo Guiotto: For example, this is example… 1 for 15.
09:11:100Paolo Guiotto: We have a limiting 3… well, let me first… let's do two examples of spicy things. Let's do first the 13, which is an example, in two variables, which is a limit…
09:25:420Paolo Guiotto: For XY going to infinity in 2 of X power 4 plus Y power 4 minus XY.
09:40:970Paolo Guiotto: Of course, now, the strategy is always the same, but the technicalities are much harder.
09:47:750Paolo Guiotto: So this problem is not straightforward. You just plug forward coordinates and you… you get out immediately. Let's see why. So this is our F function of two variables.
10:01:70Paolo Guiotto: So, F is, defined… on D, which is, in this case, R2.
10:10:20Paolo Guiotto: You can see the section, for example, FX0 immediately is X power 4 that goes to plus infinity when modulus of X goes to plus infinity, and I remind you that this means point, X0 goes to infinity.
10:27:240Paolo Guiotto: So if we have a limit, this limit must be plus infinity.
10:31:40Paolo Guiotto: Now, we should bet. Should we say that this limit does not exist, or should we prove that the limit exists and it is equal to plus infinity? This is the alternative.
10:44:200Paolo Guiotto: Now, for the first one, I should be able to see sections.
10:48:220Paolo Guiotto: Now, I know that, I have to be careful on this, I know that XY going to infinity does not mean that X and Y are going to infinity. They cannot even have a limit, so you have to be careful.
11:01:420Paolo Guiotto: And X power 4 plus Y power 4 is not X squared plus Y squared squared, for example, no? Because there is the double product, so that's not directly related to the norm.
11:15:480Paolo Guiotto: But you may expect that it has some flavor of X squared plus Y squared, so I would expect that the quantity X power 4 plus Y power 4 will go to infinity.
11:28:630Paolo Guiotto: Okay?
11:29:980Paolo Guiotto: And that is this competition with XY.
11:35:290Paolo Guiotto: So, it's difficult to see now, because you don't see raw, you… you have a fourth power minus 2nd power, so in principle, shoot the fourth power to win, but that's not always the case, so…
11:49:30Paolo Guiotto: Let's see what happens if we plug inside the polar coordinates, because at this point, I don't have any idea. So, in polar coordinates, F of raw cos theta.
11:59:480Paolo Guiotto: Rosa intita.
12:01:170Paolo Guiotto: What I show you also, there are even more tricky strategies, okay, than this one.
12:07:690Paolo Guiotto: So, with polar coordinates, I could say this becomes rho power 4 cos 4 theta plus rho power 4 sine power 4 theta.
12:18:700Paolo Guiotto: Minus XY is minus raw square cos theta sine theta.
12:24:970Paolo Guiotto: So, as you can see, we do not immediately simplify, so this is why this is a little bit more complicated.
12:32:210Paolo Guiotto: Clearly, I have rho power 4 that multiplies this coefficient cos 4th cosine theta.
12:40:300Paolo Guiotto: Sine, theta to power 4 minus raw square, costs, Theta sine theta.
12:50:940Paolo Guiotto: Okay, since it is raw to be going to infinity, let's emphasize these rows. Let's write in red.
12:58:700Paolo Guiotto: Rhaw power 4, raw power 2.
13:01:720Paolo Guiotto: So, since I see raw power 4 versus raw power tool I should say it will be raw power 4 to win, because raw is going to infinity.
13:10:750Paolo Guiotto: But I have to be careful, because if the coefficient here can be 0, then this will kill the raw power 4.
13:19:320Paolo Guiotto: I could say, is it possible to be zero? No, because to be zero, both cos and sine should be zero, and we have seen that this is impossible.
13:28:130Paolo Guiotto: Yes, but what if this quantity, I don't know, when theta varies from 0 to pi, maybe it is never zero, but it can get close to zero. We remind this argument, we have already met this type of thing, no?
13:43:930Paolo Guiotto: So that's a similar idea, so I can reuse that idea we had. Let's look at this coefficient, and let's prove that it is positive, clearly. It is never zero, clearly, but we need more. We want that to be bounded away from zero.
14:02:290Paolo Guiotto: So, be larger than a positive constant. In this way, I know that here, at least, there is positive constant times rho power 4. Remind that if the goal is plus infinite limit.
14:16:30Paolo Guiotto: The… we… what we need is a lower bound. The upper bound is easy but useless here.
14:23:550Paolo Guiotto: So, we have to put together these things with what also we have already seen. So, let's take this, this object here, let's call it,
14:35:50Paolo Guiotto: I don't know, let's use H of theta is a function. Now, H is certainly a continuous function on the interval 02 pi.
14:45:320Paolo Guiotto: well, gita ranges. I could even found the minimum value, okay, exactly. It's just a matter of computing the derivative, looking where it is zero, etc. But we don't need to do that.
15:00:260Paolo Guiotto: So, definitely there existed theta mean.
15:06:190Paolo Guiotto: In the interval 0, 2 pi.
15:09:540Paolo Guiotto: such that H of theta is greater or equal than that H of theta min.
15:15:870Paolo Guiotto: That's because of the bias thrust.
15:21:660Paolo Guiotto: TRA.
15:23:690Paolo Guiotto: And now, I know. Let's call this number, I don't know, K, as a constant.
15:31:490Paolo Guiotto: That constant is necessarily positive.
15:35:650Paolo Guiotto: Because otherwise.
15:41:430Paolo Guiotto: if K…
15:43:450Paolo Guiotto: is 0, it would mean that H of this theta min would be 0, but this would mean that you have a theta mean for which cosine power 4 of this theta mean plus
15:56:920Paolo Guiotto: The fourth power sine tetamine, would be zero.
16:01:680Paolo Guiotto: And this would say that both costs, itamin, And sign. Damina…
16:11:500Paolo Guiotto: Should be zero, and that's impossible.
16:16:560Paolo Guiotto: Okay, so now we know that, huh?
16:19:440Paolo Guiotto: Even if the coefficient depends on theta, we can throw away theta with this constant saying that this is greater than K, which is strictly positive for every theta.
16:34:790Paolo Guiotto: Okay, so I can say, now this was F,
16:38:480Paolo Guiotto: So, this allows to say that F…
16:42:820Paolo Guiotto: What happens when I replace this by the constant?
16:47:770Paolo Guiotto: I'm putting a smaller number, I'm multiplying by a positive, this quantity decreases, okay? So this means that my F, which is equal to rho power 4, etc, will be larger than
17:03:130Paolo Guiotto: raw power 4, Times.
17:06:460Paolo Guiotto: my constant k minus 9. Let's discuss about this rho power 3, and then we have, what is that?
17:15:410Paolo Guiotto: Cost assigned, okay?
17:18:230Paolo Guiotto: But this second can be treated in a rough way, because here we do not need to be such refined as for the first one, because this is just the coefficient of raw cube.
17:29:690Paolo Guiotto: We can say it is just bounded by a constant. Remind the goal is to find a G of rho here.
17:38:120Paolo Guiotto: Okay, so the problem is to find this one. So I have to throw away also this, but this can be done easily.
17:45:610Paolo Guiotto: Because what is the worst case here is that I subtract raw cube. The smallest value is when I subtract the biggest value.
17:57:180Paolo Guiotto: The biggest value I can subtract here is when this coefficient is 1. Of course, this is impossible, so who cares, but at worst, I'm sure that I will not subtract more than raw cube, yeah.
18:13:910Paolo Guiotto: Is it throws square?
18:16:90Paolo Guiotto: Okay, thank you.
18:19:330Paolo Guiotto: So, this one… Yeah.
18:24:210Paolo Guiotto: Okay, so…
18:26:240Paolo Guiotto: This means that if I say now that this is less or equal 1, why am I interested in less or equal? Because there is a minus here.
18:36:990Paolo Guiotto: So with the minus becomes greater or equal. So this becomes greater or equal than raw power 4 times constant k minus raw square. And that's my G of rho.
18:51:170Paolo Guiotto: And that's nice, because I know that K is positive, Seen some.
18:57:910Paolo Guiotto: this K is positive. G of rho goes to plus infinity when rho goes to plus infinity.
19:08:130Paolo Guiotto: Okay?
19:10:490Paolo Guiotto: you see that I don't do the upper bound, because it's useless here.
19:15:580Paolo Guiotto: Okay? I could do the upper bound if you want.
19:21:110Paolo Guiotto: upper bound…
19:27:120Paolo Guiotto: useless.
19:30:780Paolo Guiotto: is easy, and you see what is the difference, because to do the upper bound is easy, but useless.
19:36:510Paolo Guiotto: Well, where I need the lower bound, that's difficult.
19:41:730Paolo Guiotto: You see? The upper bound is easy, because F is rho power 4, then we have that cos 4 theta plus sine 4 theta minus rho square cos theta sine theta.
19:55:940Paolo Guiotto: The upper bound is easy because cos power 4 theta is less than 1, this is less than 1, so this will be less than 2 rho power 4. And then, that minus…
20:06:890Paolo Guiotto: So it means, what can be the biggest possible value here?
20:11:870Paolo Guiotto: Don't be, don't be, let's say, don't be… Cost Hussein.
20:26:850Paolo Guiotto: No, I don't have the word. However, be careful, because there is the minus, so to get the biggest thing here, you have to put the negative value here, the most negative value here.
20:38:930Paolo Guiotto: Minus, minus will become plus. So the maximum possible value is when this product is minus 1. Of course, also, this is impossible, but who cares? So we can say that this is at most, plus raw square.
20:52:750Paolo Guiotto: So, I told you that the upper bound is useless, because I know that F is less than someone who goes to plus infinity, and this does not say that F is going to plus infinity.
21:04:50Paolo Guiotto: Okay? It says the other one goes to plus infinity, but you
21:09:170Paolo Guiotto: You need the lower bound for that, okay? And that's normally, it's harder.
21:15:580Paolo Guiotto: However, we succeeded in this, and we have the conclusion. So the conclusion is… limiter.
21:23:290Paolo Guiotto: For XY, Going to infinity.
21:27:40Paolo Guiotto: of F equal plus infinity.
21:30:320Paolo Guiotto: And this is sufficient.
21:32:500Paolo Guiotto: Now, I told you that there could have been another tricky way to approach this.
21:39:370Paolo Guiotto: Right? Let me see if it is… what time we have.
21:42:370Paolo Guiotto: See, we have still… Some time, no, what is it?
21:51:370Paolo Guiotto: Let me see if, well, let's see. I, I, I, I, alternative.
22:00:780Paolo Guiotto: way.
22:03:940Paolo Guiotto: So one would say, let's look at FXY. It is X power 4 plus Y power 4 minus XY.
22:18:100Paolo Guiotto: So, the idea is, why do I want to transform this guy into X squared plus Y squared? Of course, we cannot do with X and Y, but we could say, let's call U equal X squared and V equal Y squared.
22:33:670Paolo Guiotto: It's sort of change of variables, okay?
22:37:170Paolo Guiotto: Now, this quantity becomes…
22:39:680Paolo Guiotto: U square plus V square minus X times Y. Now, here it… you have to be a little bit more ca… a little bit careful, because with this change of… it's not exactly a change of variable, because,
22:54:710Paolo Guiotto: you have the two values of X go into the same value of U, no? 1 and minus 1 give the same X square. However, let's be a little bit rough here. So this would be root of U times root of V.
23:11:110Paolo Guiotto: Now, the question is, can I say that the limit for XY going to infinity in the plane of F is equal to the limit in… well, this is another function, so let's call it, I don't know, F tilde UV.
23:28:890Paolo Guiotto: I cannot use F. F…
23:31:260Paolo Guiotto: is defined with that formula. This is another formula, okay? Can I say that this is the limit in UV going to infinity of this function, F tilde, UV?
23:45:620Paolo Guiotto: If I could say this, now it would be easier to compute that second limit, because I could use polar coordinates. Now, forget that U and V are these coordinates, but if someone gives you this problem, compute the limit of this.
24:03:770Paolo Guiotto: with the letter X and Y, you immediately would say, let's say X equals rock cos theta, Y equals Rosaine theta. So why? I cannot say, let's U equal raw cos theta and V raw sine theta. That's well possible. Letters are letters. So in this case.
24:20:250Paolo Guiotto: I would say that F tilde in rock cos theta
24:25:130Paolo Guiotto: cross and theta. These are the polar coordinates for UV0, not for X and Y, clearly.
24:30:860Paolo Guiotto: This would be raw square minus…
24:34:270Paolo Guiotto: there, we would have a raw root of cos theta sine theta. Now, you may wonder about the sign of this, because they are negative. Well, actually, here U and V are positive, so…
24:46:520Paolo Guiotto: sine theta and cosine theta should be positive. But in any case, if you look at this, this is much easier, because if you want a lower bound, this is greater than rho square minus rho. So clearly, it goes to infinity.
24:59:280Paolo Guiotto: So this would be an easier road. There is a unique problem is, can I do this change of variables?
25:05:790Paolo Guiotto: I have to convince you that XY goes to infinity if and if UV goes to infinity. So, XY goes to infinity.
25:18:440Paolo Guiotto: If and only if the norm of XY goes to infin- to plus infinity.
25:25:960Paolo Guiotto: So the norm is the root of X squared plus Y squared, or this means X squared plus Y squared goes to plus infinity, because if root goes to infinity, also the argument must go to infinity.
25:39:230Paolo Guiotto: Now, the question… the question is, is that equivalent to, say, u squared plus V square goes to plus infinity?
25:48:430Paolo Guiotto: Because, with the change of variable.
25:52:510Paolo Guiotto: remind that the change of variable is u equal X squared, V equals Y squared. So, when I say X squared plus Y squared goes to plus infinity, I'm saying U plus V goes to plus infinity.
26:06:40Paolo Guiotto: If U plus V goes to plus infinity, can I say that u squared plus B squared goes to plus infinity?
26:13:70Paolo Guiotto: UMV here, since they are squares, they must be positive.
26:22:460Paolo Guiotto: And how can I say that?
26:32:280Paolo Guiotto: That's, that's the, the problem.
26:38:110Paolo Guiotto: No, you cannot square, because if you square, you have U plus V. Yes, U plus V squared goes to plus infinity, but this is U squared plus V squared plus 2UV.
26:51:390Paolo Guiotto: Oh, maybe we can say this, yes, because this is positive.
26:55:720Paolo Guiotto: In our setup, remind that U and V are X squared and Y squared, so this is a positive.
27:05:230Paolo Guiotto: No, you see, this goes to… because this is definitely larger than U square plus V square.
27:15:460Paolo Guiotto: Yes, but I have something which is bigger is going to infinity. I don't know if something which is smaller is going to infinity, so I needed the vice versa.
27:33:490Paolo Guiotto: Nope.
27:38:790Paolo Guiotto: Because there is also an elementary to UV less or equal than U square plus V squared.
27:48:530Paolo Guiotto: Yeah.
27:50:170Paolo Guiotto: Yes, this, you know, we already seen this. This comes from U minus V squared positive.
27:57:550Paolo Guiotto: If I have this, yes, I can say that this is less than UV square plus this, which is less than this, so 2 times U squared plus V square is larger than
28:12:350Paolo Guiotto: Third…
28:13:580Paolo Guiotto: So you see that 2 times this is going to infinity, and therefore u squared plus B squared is going to infinity. So it's a bit tricky, however. I think it's a bit more,
28:25:720Paolo Guiotto: complicated than that one. So my idea was, I'm trying to simplify the function, not because I change variable with polar, but use other coordinates
28:36:400Paolo Guiotto: for which I can represent that X fourth plus Y fourth into X squared plus Y squared.
28:41:930Paolo Guiotto: To do that, I introduce this new U and V, respectively, u equal X squared, V equal Y squared. In this way, that function will be transforming that one.
28:52:390Paolo Guiotto: And I hope that my limit, when XY goes to infinity, becomes a limit when UV goes to infinity.
28:59:280Paolo Guiotto: If this is correct, it is easy to handle this one, with usual polar coordinates for the UV coordinates.
29:06:840Paolo Guiotto: So, it's like, if you look at this, this is a new problem, forget all the initial problem. Once you know that they are the same, you can forget of that, and you can focus on this one. On this one, you can use a polar coordinate for UV,
29:20:140Paolo Guiotto: There is a space of UV, you use u equal rho cosine theta, V equal rho sine theta. For that case, you get this, and easily, you have that this is going to plus infinity.
29:33:330Paolo Guiotto: with simple methods. Now, the question is, can we say that the two limits are the same? This happens if and only if, whenever you have that this goes to infinity.
29:43:180Paolo Guiotto: U of V must go to infinity, and vice versa.
29:46:70Paolo Guiotto: No? So, here we discussed this, saying that XY goes to infinity if and only if the norm of XY goes to plus infinity. This means the norm
29:57:390Paolo Guiotto: So, root of X squared plus Y squared goes to plus infinity, or equivalently, you do the square. X squared plus Y squared goes to infinity, no? Now, the question is, is this equivalent to this?
30:09:930Paolo Guiotto: Well, so the question is, what is the relation between these two? Because this one, since I put X equals X squared, V… sorry, U equals X squared, V this, this means this.
30:24:230Paolo Guiotto: U plus V goes to plus infinity. Now, can I deduce this, from this data?
30:32:860Paolo Guiotto: Well, the idea should be, okay, let's do the square.
30:36:170Paolo Guiotto: U plus B squared goes to plus infinity. If you do the square, we have, developing the square, we have this.
30:42:770Paolo Guiotto: We may notice that our U and V are positive, they are squares. This time could be eliminated, but I would have that this is larger than this.
30:53:950Paolo Guiotto: And this is not the good argument, because yeah, if the bigger goes to plus infinity, what can be said about the smaller?
31:00:530Paolo Guiotto: I don't know. I need to show that this can be made bigger than this.
31:06:370Paolo Guiotto: Here I use this simple inequality. However, this argument is not elementary, as you understand, because it's not immediate that you have this kind of idea.
31:18:120Paolo Guiotto: So you say 2UV is less than U squared plus V squared, that comes from developing this square. So I can say that this thing is less or equal than U square plus V squared.
31:28:440Paolo Guiotto: plus this replaced by U squared plus V squared. So, 2 times U squared plus V squared.
31:35:530Paolo Guiotto: So this one is greater than this one that goes to plus infinity, so also this goes to plus infinity.
31:41:880Paolo Guiotto: And since this is 2 times this, this one must go to plus infinity, okay?
31:47:220Paolo Guiotto: So at the end, we can say that the two limits are the same.
31:52:540Paolo Guiotto: Good. Well, now time is over, and so, I would say,
32:00:960Paolo Guiotto: Well, do the exercises, left.
32:05:520Paolo Guiotto: And, which are due.
32:14:450Paolo Guiotto: The 1.8.10, these are all limits at infinity.
32:21:50Paolo Guiotto: Be careful, because some of them can be… can have difficulties of the kind we have seen in this example. If you want, you can try to read by yourself the example 1415.
32:37:520Paolo Guiotto: study… Example 1.4.15.
32:46:450Paolo Guiotto: However, we will compute limits along the course many, many times, so it's not something that we finish now, and we never will compute anymore.
32:56:420Paolo Guiotto: Okay, thank you, have a nice day, and have a nice weekend. Now, what's happening here…
33:06:510Paolo Guiotto: Why…