<s>[INST] How do I show that this Limit of 2 variables is zero? How do I show that :$$\lim_{(x,y)\to(0,0)}xy\frac{x^2-y^2}{x^2+y^2}=0?$$
I'm stumped...
 [/INST] Personally, I would take a change of coordinates and look at the limit in polar coordinates.
Substituting $x = r\cos\theta$ and $y = r\sin\theta$, we get
\begin{array}
xxy \frac{x^2-y^2}{x^2+y^2} &=& r^2\cos\theta\sin\theta\frac{r^2\cos^2\theta-r^2\sin^2\theta}{r^2\cos^2\theta + r^2\sin^2\theta} \\ \\
&\equiv& r^2(\cos^3\theta\sin\theta - \cos\theta\sin^3\theta) \\ \\
&\equiv& \frac{r^2}{4}\sin 4\theta
\end{array}
As $r \to 0$, this tends towards 0 for all values of $\theta$.
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