Find the following limit : $\lim_{(x,y)\rightarrow(0,0)}xy \log\left|y\right|$

Question

I have been trying to solve this limit: $$\lim_{(x,y)\rightarrow(0,0)}xy\log\left|y\right|$$

I tried using polar coordinantes but i could not find the answer. I tried using iterate limits but i dont get any information . Any hint on how to solve this?

Answer 1

Recall that as $t\to 0^+$

$$t\log t\to 0$$

then

$$xy\log\left|y\right|=x\cdot \left(y\log\left|y\right|\right)\to 0\cdot 0=0$$

Answer 2

Hint. Note that $|y\ln|y||\leq e^{-1}$ for $0<|y|\leq 1$.