<s>[INST] Show $\lim\limits_{(x,y)\to(0,0)} \frac{xy}{\sqrt{x^2+y^2}}=0$ using Delta Epsilon Proof I tried following this answer (https://math.stackexchange.com/a/542303/86425), but I cannot see how this shows the limit is 0.  Can someone please provide guidance
 [/INST] Note that $\sqrt{x^2 + y^2} \geq \sqrt{y^2} = |y|$.  Then
$$
\left|\frac{xy}{\sqrt{x^2 + y^2}}\right| = |x|\frac{|y|}{\sqrt{x^2 + y^2}} \leq |x|
$$
so
$$
\lim_{(x,y) \to (0,0)}\left|\frac{xy}{\sqrt{x^2 + y^2}}\right| \leq \lim_{(x,y) \to (0,0)} |x| = 0 \, .
$$
Personally, I prefer thinking of these problems in polar coordinates.  So let $x = r \cos\theta$, $y = r \sin \theta$ where $r = \sqrt{x^2 + y^2}$.  Then
\begin{align*}
\lim_{(x,y) \to (0,0)} \frac{xy}{\sqrt{x^2 + y^2}} &= \lim_{r \to 0} \frac{r^2 \cos \theta \sin \theta}{r} = \lim_{r \to 0} r \cos \theta \sin \theta = 0 \, .
\end{align*}
</s>