Can't prove or disprove the existence of a limit

Question

I can't prove or disprove the existence of $\lim_{(x,y)\to (0,0)}\frac{x^3+y^2}{x^2+y}$. I have already tried several paths but still I can't prove that the limit doesn't exist. Any ideas?

Answer 1

Let $y=x^{4}-x^{2}$, then \begin{align*} \lim_{x\rightarrow 0^{+}}\dfrac{x^{3}+(x^{4}-x^{2})^{2}}{x^{2}+x^{4}-x^{2}}=\lim_{x\rightarrow 0^{+}}\dfrac{x^{8}-2x^{6}+x^{4}+x^{3}}{x^{4}}=\infty. \end{align*}