I want to calculate the limit
$$\lim_{(x, y)\to(0, 0)}x y e^{1/\sqrt{x^2 + y^2}}$$
So far I have tried substitution y=kx, but that gave me result $0 * e^∞$ as it is indefinite form. I have no result. I have also tried polar coordinates $x=r \cos x, y= r \sin x$ and as result I got $\lim_{r \to 0}r^2 (\sin x \cos x) e^{1/r}$. Only idea I got from that is L'hopital's rule, but I have no idea how many times I need to apply it to see result. Using Wolfram Alpha I got that result is 0. Am I on right way, and what to do next ( how many times is enough to apply L'hopital's)? Every help is appreciated.