Nonexistence of the limit $\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y^2}{x^3+y^3}$ How can we prove that the limit $\lim_{(x,y)\rightarrow (0,0)} \dfrac{x^2y^2}{x^3+y^3}$ doesn't exist?
I have tried a lot of different paths and all of them lead to zero. I have only tried paths belonging to the domain and so I thank you for the lights that you've thrown here for me. When I've plotted the graphic of the function, it was clear that this limit doesn't exist, but could not prove it. For instance, this is an exercise of the book of Louis Leithold.

A good way is to find a path through the plane $(x,y)$ such that the limit is different from the limit from another path.
For exemple, following the path $y=x$ we have the limit of $0$. If you find a path with a different limit, then the limit doesn't exist.
