I'm solving this for help a freshman student in physics but I'm thinking that I don't know the right way to think about this.
The problem is solve that limit (if this exists):
$$ \lim_{(x,y)\rightarrow(0,0)}\frac{xy}{y-x^3} $$
My first attempt was check if $\frac{x}{y-x^3}$ or $\frac{y}{y-x^3}$ is limited. But those functions are not limited. So, my second attempt was make that change: $x=r\cos\theta$ and $y=r\sin\theta$ and view what happens when $r\rightarrow 0$
$$ \frac{r^2\cos\theta\sin\theta}{(r)[\sin\theta-r^2\cos\theta]}=\frac{r\cos\theta\sin\theta}{\sin\theta-r^2\cos\theta}\rightarrow 0 \textrm{ when } r\rightarrow 0 $$
So the limit is $0$! But that looks like wrong and this is confusing me. If I calcule this limit in the "Wolfram Alpha" the results is $0$, but if I calcule that over the curve $(^{3}\sqrt{t-t^2},t)$ that show us the limit doesn't exist (because over that curve the function goes to infinity).
The question is: What I made wrong? Why?