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?

Yes it is correct since


*

*$x=0 \implies \frac{x^3}{y^2}e^{\frac{-x^2}{y}}=0$

*$y=x^2 \quad x\to 0^+ \implies \frac{x^3}{y^2}e^{\frac{-x^2}{y}}=\frac{1}{xe}\to +\infty$
the limit doesn't exist.
