$$f(x,y)=\begin{cases} y- \frac{e^{x^2+y^2}-x^2-y^2}{x^2+y^2},& x^2+y^2 \neq 0. \\ -1, & x=y=0 \end{cases}$$
I keep runnung into trouble with these types of questions. The way I do them is the same as in this question: Prove that the following functions is differentiable on $(-1,1) \times \mathbb R$
But I run into trouble with the limits.. I was thinking polar coordinates, but am no sure how do put it together.