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?

Question

$\lim\limits_{(x,y) \to (0,0)}\ {xy\log(x^2+y^2)}$

I tried this:

$$xy\log(x^2+y^2)=xy\log(x^2)+xy\log(1+y^2/x^2)$$

$|xy\log(x^2)|=2|x||y||\log|x||$ (because I use $(x,y)\neq(0,0)$) $\le 2|x||y|||x|-1|$ and taking the limit, $\lim_{(x,y) \to (0,0)}\ {xy\log(x^2)}=0$.

Since the other member has two variables in the argument of $\log$, I don't know how to continue. I know the limit of this function is zero because proceeding with other methods I've seen the limit is zero.

Answer 1

Hint. Note that $|xy|\leq (x^2+y^2)/2$ (expand $(|x|-|y|)^2\geq 0$) and $$\lim_{t\to 0^+} t\cdot \log t =\lim_{s\to +\infty} \frac{-s}{e^{s}}=0 \quad\mbox{(where $s=-\log t$)}.$$