TITLE: Finding $\lim_{n \to \infty} n(\sqrt[n]{2} - 1)$
QUESTION [9 upvotes]: I am trying to find the limit of:
$$\lim_{n \to \infty} n(\sqrt[n]{2} - 1)$$
I know it should be very simple but I don't seem to get it.
Thanks in advance for any help!

REPLY [4 votes]: There are several answers and comments suggesting L'Hospital's
rule.
In fact, it is a bit easier - you only need the definition of the
derivative.
$$\lim\limits_{n\to\infty} \frac{2^{1/n}-1}{1/n}= \lim\limits_{x\to0} \frac{2^x-2^0}{x-0}$$
So this expression is precisely the value of the derivative of
$f(x)=2^x$ at $x=0$.
If you already know that the derivative of $f(x)=2^x$ is
$f'(x)= 2^x  \ln 2$, then you have
$$\lim\limits_{n\to\infty} \frac{2^{1/n}-1}{1/n}= f'(0)=\ln 2.$$