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.

I am going to essentially repeat nbubsis's answer, but with a little more explanation.  A lot of people (including me and probably nbubis) would say that the limit does not exist because there does not exist $r>0$ such that the function $x^2y^2/(x^3+y^3)$ is well-defined in a "deleted neighborhood" of $(0,0)$ of the form $\{(x,y) \mid 0< \sqrt{x^2+y^2}<r\}$ (nbubis explains why).  Other people would say you only have to consider pairs $(x,y)$ that belong to the domain of the function $f(x,y)=x^2y^2/(x^3+y^3)$.  This seems to be what Victor Chaves and kigen  are suggesting.  I'm not sure whether either interpretation is universally accepted as correct.  My interpretation seems to trivialize the problem, but it is a popular one.  The "epsilon-delta-neighborhood" definition that Steven refers to seems to favor my interpretation as well.  I suggest you ask your teacher (I can't imagine anyone doing this problem for fun - is this homework?) which interpretation they want, and do it the way s/he wants.
EDIT: it might turn out in the end that it doesn't matter which of the two interpretations you use.  But even if this happens, the issue I describe above might be relevant to some other problem, and I suggest you ask your teacher how you should interpret such a question.  
