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I'm trying to compute this limit $\lim_{(x,y) \to (0,0)}2x\sin^2(\frac{1}{y})$, but WolframAlpha says that it does not exist.

I'm not quite sure why. I do understand that there are oscillations coming from the $\sin(1/y)$. However, $x \to 0$ as well. Shouldn't that crush the function to zero?

Also, I know that $\lim_{x \to 0} x \sin(\frac{1}{x}) = 0$. Isn't that pretty much the same idea as the limit in question?

Siong Thye Goh
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user1691278
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    The function isn't defined on the $x$-axis. If you don't worry about that, then $|f(x,y)|\le 2|x|$ and so tends to zero, as you say. With Wolfie A you get what you pay for. – Angina Seng Feb 02 '18 at 06:54
  • Why is it not defined on the x-axis? – user1691278 Feb 02 '18 at 07:00
  • @LordSharktheUnknown I see what you mean. How can I not worry about that though? – user1691278 Feb 02 '18 at 07:05
  • It is not defined on the $x$-axis because on the $x$-axis, $y=0$. – Mitchell Faas Feb 02 '18 at 07:07
  • @MitchellFaas So you're saying when the limit approaches from the $y$ axis to zero first, it is undefined? – user1691278 Feb 02 '18 at 07:09
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    @user1691278 No, that would be defined. The line that lies exactly on the $x$ axis, and only that line is undefined. So if we force $y=0$, things don't work. If you don't care about that particular path, then the limit does exist. i.e the limit is $0$ for every path except the one on the $x$ axis. – Mitchell Faas Feb 02 '18 at 07:13
  • @MitchellFaas How can we force $y = 0$? We are computing the limit as $(x,y) \to (0,0)$? I thought it was totally fine that the function is not defined at $y = 0$ - isn't that why we're computing the limit in the first place? – user1691278 Feb 02 '18 at 07:15
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    @user1691278 In the 1-D case this is indeed what happens, but try to think of it in terms of distances and points. In 1-D space: Suppose we pick a point $p$ ($0$ in this case) and a variable $x$. We wish to compute the limit when the distance $d(p,x)$ between $p$ and $x$ tends to $0$ (but does not actually attain it). In the 2-D space we do the same: We pick a point $p, ((0,0))$ in this case) and compute the limit as $d((0,0),(x,y))$ tends to $0$. But if we choose $x=1, y=0$ we still have a non-zero distance: $d((0,0), (1,0)) = 1$. So we can force $y=0$ and still compute the limit. – Mitchell Faas Feb 02 '18 at 07:24
  • @MitchellFaas Brilliant explanation! – user1691278 Feb 02 '18 at 07:26
  • As a little side-note: Since we're only concerned with distance you can take any path (it doesn't have to be a straight line, but could for example also be a spiral). – Mitchell Faas Feb 02 '18 at 08:12

2 Answers2

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It is equal to $0$ according to Wolfram alpha.

enter image description here

The domain of the function should exclude the $x$-axis, that is the domain $D = \{ (x,y) : y \neq 0\}$.

Let $\epsilon > 0$, we choose $\delta = \frac{\epsilon}2$, if $(x,y) \in D$ and $\sqrt{(x-0)^2+(y-0)^2} < \delta$

then

$$ \left|2x\sin^2 \left( \frac1y\right)-0\right| = \left|2x\sin^2 \left( \frac1y\right)\right|\leq 2|x| \leq 2 \delta < \epsilon.$$

Siong Thye Goh
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  • I don't have the pro version, but I see this: http://www.wolframalpha.com/input/?i=lim+2xsin%5E2(%5Cfrac%7B1%7D%7By%7D)+as+(x,y)+-%3E+(0,0) – user1691278 Feb 02 '18 at 07:01
  • that's interesting thing that wolfram alpha gives different answer according to versions. – Siong Thye Goh Feb 02 '18 at 07:02
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    I presume one has to pay for the Pro version. – Angina Seng Feb 02 '18 at 07:05
  • @LordSharktheUnknown How can you have two answers though? – user1691278 Feb 02 '18 at 07:06
  • @SiongThyeGoh The problem is that with the non-pro version, the allotted computational time gets exceeded. So wolfram just says it doesn't exist, adding that it may be dependent upon the path in complex space. i.e It hasn't been able to find the answer in a small enough time frame. – Mitchell Faas Feb 02 '18 at 07:10
  • @user1691278 The free WA does indeed return limit does not exist, but note the box at the bottom that says "Standard computation time exceeded..." and the popup next to it "Extended Computation Time: As a Wolfram|Alpha Pro subscriber, you can request triple the computation time from the Wolfram|Alpha servers allowing for additional or more detailed results.". It's not that the Pro subscription buys you a different math, but it does buy you more time for WA to figure the right math. – dxiv Feb 02 '18 at 07:12
  • @SiongThyeGoh I see. However, I still don't understand why the limit being undefined at $y=0$ is an issue. Isn't that why we compute limits in the first place? Even if it's undefined, as long as we can find a point such that we can bind it epsilon close? – user1691278 Feb 02 '18 at 07:21
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$$|2x\sin^2(1/y)|\le 2|x|\cdot |1|\le 2|x|.$$

Since $2|x|\to 0$ as $x\to 0$, the limit is not undefined.

Asim90
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  • Doesn't this contradict @MitchellFaas's comment? – user1691278 Feb 02 '18 at 14:30
  • @user1691278 idk what coment he made – Asim90 Feb 03 '18 at 05:25
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    @user1691278 It does not :) The limit is defined as long as $y\neq 0$. The inequality Asim90 writes no longer holds in this case: Say $y = 0$ Then $1/y$ is undefined, so $2x\sin^2(1/y)$ is undefined, so we do not know its relationship with the right part of the equality. This has to do with the domain of $\sin(x)$. Think of it like this: Does the inequality $|\sin({monkey})|\leq1$ make any sense? If that doesn't, then surely $|\sin({Undefined})|\leq1$ makes no sense. – Mitchell Faas Feb 03 '18 at 18:55