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.

An approach similar to G Tony Jacobs:  use the continuity of logarithm (i.e. $\log \lim f(x) = \lim \log f(x)$) to log the expression to get 
$$
L f(x) = \frac{\log x}{x}
$$
then show it converges to $0$ by L'Hospital's rule, then exponentiate back to get 1. 
