Help to evaluate this limit $\lim_{x \to \infty}x^{\frac{1}{x}}$

Question

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.

Answer 1

Hint

Use the L'Hôpital's rule to find

$$\lim_{x\to\infty}\frac{\ln x}{x}$$

Answer 2

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?

Answer 3

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.