An unknown limit with nth root: $\lim\limits_{n\to\infty}n(x^{1/n}-1)$

Question

How can I find such a limit: $$\lim_{n\to\infty}n(x^{1/n}-1)$$ I tried using a kind of binomial formulas. But nothing helped so far.

Thank you!

Answer 1

Write the limit as $n(x^{1/n}-1) = \frac{(x^{1/n}-1)}{1/n}$, make substitution $t= 1/n$, this is the definition of the derivative at zero for $d(x^t)/dt$ $$ \lim_{t\to 0^+}\frac{(x^{t}-x^0)}{t-0} = \frac{dx^t}{dt} $$

Answer 2

Hint: We have $x^{1/n}=e^{\log x/n}$. Now use the first two terms of the Maclaurin series of $e^t$.