Show that the limit in $(0,0)$ of $\frac{xy}{x^3-y}$ doesn’t exist.

Question

I'm trying to show that the limit below doesn't exist, I think it's quite simples but I couldn't until now.

$$\lim_{(x,y)\to (0,0)}\dfrac{xy}{x^3-y}$$

My approach is try to find two curves in the domain that have different limits in $(0,0)$

I couldn't find any curve that the limit isn't $0$.

Any tips? Thanks.

Answer 1

Your idea is fine. Take the curve $y=x^3+x^4$. Then$$f(x,x^3+x^4)=-x-1$$and the fact that $\lim_{x\to0}-x-1=-1$. On the other hand, $\lim_{x\to0}f(x,0)=\lim_{x\to0}0=0$.