[Math] Limit of a multivariable function: $x^2\log(x^2+y^2)$, $(x,y)\to(0,0)$

Question

Can someone help me to solve this limit?

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

For any line $y=mx$ the result is $0$, so, the candidate is $0$.
I tried to use the squeeze theorem, polar coordinates, etc. but I can't solve it.

Thanks.

Answer 1

$0\le|x^2\log(x^2+y^2)|\le|(x^2+y^2)\log(x^2+y^2)|$. But $t\log(t)\to0$ when $t\to 0$