Trying to show that $\ln(x) = \lim_{n\to\infty} n(x^{1/n} -1)$ How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?
I ran into this identity on this stackoverflow question. I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been a failure.

$\lim_{n\to\infty}n(x^{\frac{1}{n}}-1)=\lim_{n\to\infty}\frac{x^{\frac{1}{n}}-1}{\frac{1}{n}}=f^{\prime}(0)$, where $f(t)=x^t$. Since
$$ f^{\prime}(t)=\ln(x)x^t$$
it follows that $f^{\prime}(0)=\ln(x)$.
