Partial derivative and limit.

Question

I have problem with the limit : $$\lim_{(h,k) \rightarrow (0,0)} \frac{hk+h|h|}{(|h|+|k|)\sqrt{h^2+k^2}}.$$ I tried to use polar coordinates. I am not sure what to do with the absolute value.

Answer 1

If $k=0$, the limit is $0$. And if $h=k>0$, then the limit is $\frac1{\sqrt2}$. Therefore, the limit does not exist.

Answer 2

use that $$|h|+|k|\geq 2\sqrt{|h||k|}$$ and $$\sqrt{h^2+k^2}\geq \sqrt{2|h||k|}$$