Limit of a function with two variables existance I have the function $f(x,y)=\frac{x^3}{y^2}e^{\frac{-x^2}{y}}$, and I have to show whether the following limit exists: $\lim\limits_{(x,y)\to(0,0)}f(x,y)$.  If I set $y=x$, then $\lim\limits_{(x,x)\to(0,0)}f(x,x)=\lim\limits_{x \to 0}f(x,x)=0$; and if I set $y=x^2$ the limit does not exist. Is this enough to say that the $\lim\limits_{(x,y)\to(0,0)}f(x,y)$ does not exist?

If the limit exists, let it be $c$. Then for a given $\epsilon > 0$ there exists $\delta > 0$ such that $|f(x,y)-c|<\epsilon \quad \forall \quad |x|<\delta,|y|<\delta$.
Hence, taking $y=x^2$ in particular, we can choose $0<\delta_1<\delta$ such that $|x|<\delta_1 \implies x^2<\delta$.
Thus we have, $|f(x,x^2)-c|<\epsilon \quad \forall \quad |x|<\delta_1$.
Which means $\lim_{x\rightarrow 0}f(x,x^2)$ exists and this gives a contradiction.
