Evaluate limit to infinity of roots of $n$ [ $\lim\limits_{n \to \infty} \sqrt n (\sqrt[n]{n}-1)$ ]

Question

Here is the problem:

Evaluate $\lim_{n \to \infty} \sqrt n (\sqrt[n]{n}-1)$

At my disposal:

For $ n \ge 1, n \in \mathbb{N}$ , we have $(\sqrt[n]{n}-1)\le 2/ \sqrt n $

Hence , the upper bound of the expression is $2$.

I suspect that the limit is $0$.

A hint please for an elementary solution without logs.

Thank you.