The Limit of n^(1/n)
Formula
$$ \lim \limits_{n \to \infty} \sqrt[n]{n} = 1 $$
$$ \lim \limits_{n \to \infty} \sqrt[n]{\dfrac{1}{n}} = 1 $$
Proof
Instead of $\sqrt[n]{n}$, it is easier to find the limit of $\ln \sqrt[n]{n}$.
$$ \lim\limits_{n \to \infty} \ln \sqrt[n]{n} = \lim\limits_{n \to \infty} \dfrac{\ln n}{n} $$
Since it is of the form $\dfrac{\infty}{\infty}$, by L’Hôpital’s rule,
$$ \lim\limits_{n \to \infty} \dfrac{\ln n}{n} = \lim\limits_{n \to \infty} \dfrac{\dfrac{1}{n}}{1} = \lim\limits_{n \to \infty} \dfrac{1}{n} = 0 $$
Therefore,
$$ \lim\limits_{n \to \infty} \ln \sqrt[n]{n} = 0 \implies \lim\limits_{n \to \infty} \sqrt[n]{n} = 1 $$
The second equation can be proven in the same way.
■