I'm not quite sure how to approach the problem. I tried to approach the limit along the $x$ axis and the $y$ axis but I keep getting $0$. I've done the partial derivatives as well but I don't see how that helps me. Could you give me a hint on how to start it? I think I don't really understand the problem.
for your control $$\frac{\partial f(x,y)}{\partial x}=\frac{y \left(x^4+4 x^2 y^2-y^4\right)}{\left(x^2+y^2\right)^2}$$ and $$\frac{\partial f(x,y)}{\partial y}=\frac{x \left(x^4-4 x^2 y^2-y^4\right)}{\left(x^2+y^2\right)^2}$$