How to obtain the limit of $xy\log(x^2+y^2)$ when $(x,y)\to(0,0)$ without using polar coordinates or L'Hôpital?
Hint. Note that $|xy|\leq (x^2+y^2)/2$ (expand $(|x|-|y|)^2\geq 0$) and $$\lim_{t\to 0^+} t\cdot \log t =\lim_{s\to +\infty} \frac{-s}{e^{s}}=0 \quad\mbox{(where $s=-\log t$)}.$$