# Find the limit as x approaches infinity of x ^(1 /x)?

Sep 8, 2014

By using logarithmic properties,
${\lim}_{x \to \infty} {x}^{\frac{1}{x}} = 1$

Let us look at some details.
Since $x = {e}^{\ln x}$,
${\lim}_{x \to \infty} {x}^{\frac{1}{x}} = {\lim}_{x \to \infty} {e}^{\ln {x}^{\frac{1}{x}}}$
by the property $\ln {x}^{r} = r \ln x$,
$= {\lim}_{x \to \infty} {e}^{\frac{\ln x}{x}}$
by squeeze the limit in the exponent,
$= {e}^{{\lim}_{x \to \infty} \frac{\ln x}{x}}$
by l'Hopital's Rule,
${e}^{{\lim}_{x \to \infty} \frac{\frac{1}{x}}{1}} = {e}^{0} = 1$