Answer
Proof given below.
Work Step by Step
Let $L=\displaystyle \lim_{n\rightarrow\infty}(n)^{1/n}$.
$ L=\displaystyle \lim_{n\rightarrow\infty}(n)^{1/n}\qquad$ ...apply ln(..) to both sides
$\displaystyle \ln L=\ln[\lim_{n\rightarrow\infty}(n)^{1/n}]\qquad$ ... ln is continuous,
$\displaystyle \ln L=\lim_{n\rightarrow\infty}[\ln(n)^{1/n}]$
$\displaystyle \ln L=\lim_{n\rightarrow\infty}[\frac{1}{n}\ln n]$
$\displaystyle \ln L=\lim_{n\rightarrow\infty}\frac{\ln n}{n}\qquad$ ... $(\displaystyle \frac{\infty}{\infty}$, apply L'Hospital's rule)
$\displaystyle \ln L=\lim_{n\rightarrow\infty}\frac{1/n}{1}$
$\displaystyle \ln L=\lim_{n\rightarrow\infty}\frac{1}{n}$
$\ln L=0$
$L=1$
