Help to evaluate this limit $\lim_{x \to \infty}x^{\frac{1}{x}}$ What is the value of this limit?
$$
\lim_{x \to \infty}x^{\frac{1}{x}}
$$
I have never encountered such a limit before, so any help or advice would be much appreciated.

Here's a start:
$\lim x^{1/x} = \lim \exp(\log(x^{1/x)})) = \lim \exp\left[\frac1x\log x\right] = \exp\left[\lim\frac1x\log x\right]$.
The limit in that last expression is a $0\cdot\infty$ form. Do you know how to handle those with L'Hôpital's Rule?
