Is this a standard result?

$\displaystyle \displaystyle\lim_{n\to\infty}a_n^{n}=\left( \lim_{n\to\infty}a_n\right)^n$

Because, if I'm given this question:

Find

$\displaystyle \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\ right)^{4n}$

It would equal to

$\displaystyle \displaystyle\lim_{n\to\infty}\left[\left(1+\frac{1}{n}\right)^n\left(1+\frac{1}{n}\ri ght)^n\left(1+\frac{1}{n}\right)^n\left(1+\frac{1} {n}\right)^n\right]$

$\displaystyle \displaystyle\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\righ t]^4$