Limit of $f(x,y) = xy \log(x^2+y^2)$ as $(x,y) \to (0,0)$?

Question

I am trying to find whether $\lim_{\vec{x} \to \vec{0}} f(x,y) = xy \log(x^2+y^2)$ exists? I have tried several things but nothing fruitful so far, does anyone have a hint on how to solve this?

Answer 1

By polar coordinates

$$\lim_{\vec{x} \to \vec{0}} xy \log(x^2+y^2)=\lim_{r\to0^+}2r^2\cos \theta\sin\theta \log r= 0$$

indeed for $t\to 0^+$

$$t\log t \to 0$$