The n-th root of n

Idea. The base $n$ grows, while the exponent $\frac1n$ shrinks. Who wins? Taking the logarithm shows that the exponent wins.

Computation. For $n\ge1$,
$$n^{1/n}=\exp\!\left(\frac{\ln n}{n}\right).$$
We know that $\dfrac{\ln x}x\to0$ as $x\to+\infty$ (see the problem *The logarithm is slower than any power*), in particular along the integers. By continuity of $\exp$, $n^{1/n}\to e^0=1$.