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Show that

$$\lim_{(x,y)\to (0,0)} \frac{xy^3}{1+y^3} = 0$$

The only way I know of doing it is using squeeze theorem, but I couldn't find any function. Any hint?

Thanks!

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    $\begingroup$ I don't see the problem, letting either x or y tend to zero will yield 0 as your limit... $\endgroup$ – Asier Calbet Sep 28 '14 at 19:14
  • $\begingroup$ @Assaultous2 You don't get to treat $x$ and $y$ separately here. You have to consider approaching $(0,0)$ along all paths. $\endgroup$ – James Sep 28 '14 at 19:15
  • $\begingroup$ Indeed, thanks guys! $\endgroup$ – Giiovanna Sep 28 '14 at 19:17
  • $\begingroup$ @James I up voted Assaultous2's comment perhaps because I misunderstood him. What I first I understood from his comment is that the function is defined on $(0,0)$, so it's enough (depending on what the OP knows) to just replace $x$ and $y$ with $0$. $\endgroup$ – Git Gud Sep 28 '14 at 19:17
  • $\begingroup$ I suppose that reading is also valid. I was working on the assumption that the OP doesn't get to appeal to continuity. $\endgroup$ – James Sep 28 '14 at 19:21
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For every $|y|\leqslant\frac12$, $|1+y^3|\geqslant1-\left(\frac12\right)^3\geqslant\frac12$ hence $$\left|\frac{xy^3}{1+y^3}\right|\leqslant2\cdot|x|\cdot|y|^3\leqslant|x|\cdot|y|,$$ which should be enough to conclude.

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You can use polar coordinates as $$ x=r\cos\theta \ \ y=r\sin\theta $$ and the limit as $r\to 0$. The limit will be $$ \lim_{(x,y)\to (0,0)} \frac{xy^3}{1+y^3} = \lim_{r\to 0} \frac{r^4\cos\theta\sin^3\theta}{1+r^3\sin^3\theta} = \frac 01=0 $$

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    $\begingroup$ This doesn't seem to help much as the function is defined on the limit point. $\endgroup$ – Git Gud Sep 28 '14 at 19:15

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