How to prove that the limit of $f(x,y)=\frac{xy}{2(x+y)}$ at $(0,0)$ is 0 using the definition of limits?

Question

I got to the point

$|f(x,y)|=\frac{|x||y|}{2\,|x+y|} \leq \frac{x^2+y^2}{2|x|+2|y|}$

but then I'm stuck as I can't see how I could make the denominator smaller such that I create a $\sqrt{x^2+y^2}$ type expression. Any help would be appreciated.

Answer 1

But the limit is not $0$. Note that$$\lim_{x\to0}f(x,-x+x^2)=\lim_{x\to0}\frac{x-1}2=-\frac12.$$