How to prove $\lim_{(x,y) \rightarrow (0,0)} \frac{x^{2}y^{2}}{x^{3}+y^{3}}$ doesn't exist

Question

I have tried using polar coordinates and several polynomial functions, but they all converge to the value of $0$; checking at Wolfram Alpha, however, it confirms that the limit does not exist. How should I approach these kinds of questions and not get stuck? $$ \lim_{(x,y) \rightarrow (0,0)} \frac{x^{2}y^{2}}{x^{3}+y^{3}} $$

Answer 1

We have that

• $x=0 \implies f(0,y)=0$
• $ x=t \quad y=-t+t^2 \quad t\to 0$ $$\implies\frac{x^2y^2}{x^3+y^3}= \frac{t^2(t^2-2t^3+t^4)}{t^3-t^3+3t^4-3t^5+t^6}=\frac{1-2t+t^2}{3-3t+t^2}\to \frac13$$

therefore the limit doesn't exist.

Answer 2

HINT: Take, e.g., $y^3=-x^3+x^6$. The easiest way is to prove now that $$ \lim_{(x,y) \rightarrow (0,0)} \left(\frac{x^{2}y^{2}}{x^{3}+y^{3}}\right)^3 $$ does not exist.