Is there any good way to find a multivariable limit other than switching to polar coordinates?
For example, students each year are inundated with problems like $$\lim_{(x,y)\to (0,0)}{\frac{x^2y-xy^2}{\sqrt{x^2+y^2}}}$$ and, putting aside using the definition of the limit directly, the only good solution that seems to exist is to write $$\lim_{r\to 0}{\frac{r^3(\cos^2\theta\sin\theta-\cos\theta\sin^2\theta)}{r}}=\lim_{r\to 0}{\;r^2\cos\theta\sin\theta (\cos\theta -\sin\theta)}=0$$
Also, is there any instance in which this method fails? Don't feel obligated to provide a full-blown proof, I'm just wondering if there is a proof because I don't recall ever seeing one in a textbook.
The polar coordinates trick usually works nicely if the denominator is a relative of $x^2+y^2$. This happens fairly often in exercises.
There certainly are other methods. For example, let $w=\max(|x|,|y|)$. Then the denominator is $\ge w$. The numerator has absolute value $\le 2w^3$. Thus the ratio is $\le 2w^2$, which approaches $0$ as $(x,y)\to (0,0)$.