find a limit of $\lim_{n\rightarrow\infty}\left(\frac{a^{1/n}+b^{1/n}}{2}\right)^n$ $$\lim_{n\rightarrow\infty}\left(\frac{a^{1/n}+b^{1/n}}{2}\right)^n$$
No idea how to do it please help step by step

the natural log of both sides should yield an answer (after applying l'Hopitals rule).
\begin{align*}
\lim_{n \to \infty}f(n)&= \left(\frac{a^{1/n}+b^{1/n}}{2}\right)^n \\
\lim_{n \to \infty}\log n&= \frac{\log \left(\frac{a^{1/n}+b^{1/n}}{2}\right)}{1/n} \\
&=\lim_{n \to \infty}\frac{-(a^{1/n}\log(a) +b^{1/n}\log(b))/(2n^2)}{-1/n^2}\\
&=\lim_{n \to \infty}\frac{1}{2}\left(a^{1/n}\log(a) +b^{1/n}\log(b)\right)\\
&=\frac{1}{2}\log(ab)
\end{align*}
So $$\lim_{n \to \infty}f(n)=(ab)^{1/2}$$
