Consider the function
$$f(x)=x^{1/x}\tag{1}$$
defined for $x>0$.
We can write this as
$$f(x)=e^{\frac{1}{x}\log{x}}$$
Does this function have a limit when $x\to 0$? Maple tells me there isn't.
Also, $\frac{1}{x}\log{x}$ apparently doesn't have a limit when $x\to 0$ either.
My first thought was that the latter would be $-\infty$.
What happens to the function in $(1)$ near $0$?
EDIT: as has become clear in the comments, the limits are indeed very simple to compute. My doubts arose because Maple was giving me a weird result, but as per the comments, I needed to tell Maple to compute the limit from above at $0$.
So,
$$\lim\limits_{x\to 0^+} \frac{\log{x}}{x}=-\infty$$
$$\lim\limits_{x\to 0^+} x^{1/x}=0$$