Evaluate $\lim_{x\rightarrow 0}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right]$ I am interested in the following limit:
$$\lim_{x\rightarrow 0}\frac{1}{\sqrt{x}}\exp\left[-\frac{a^2}{x}\right]$$
Does this limit exist for real $a$?
Edit: I am only interested in the case when $x$ is non-negative. Thanks for reminding.

The left hand limit is not defined because of the square root in the denominator. So, you just need to check the value of the function at $f(0^+)$ and $f(0)$. If these two are equal, then limit is defined at $x=0$.
