$f(x,y)= \begin{cases} \frac{x^{2}y^{2}}{x^{3}+y^{3}}, x\ne -y \\0, x=-y \end{cases} $
I got that $\lim_{(x,y) \to (0,0)}f(x,y)=\lim_{r \to 0}r\frac{cos^{2}\phi sin^{2}\phi}{cos^{3}\phi + sin^{3}\phi}$ when switching to polar coordinaates, but I'm stuck here, since the phi part doesn't seem to be bounded, so I don't even really know where to go from here. I'm wondering if the limit doesn't exist then go along some different curves and show they have different limits, but if that's the case then I don't know which ones to even pick.