I need to find the limit of
$$\lim _{x \rightarrow \infty}\left(\frac{a^{1 / x}+b^{1 / x}}{2}\right)^x.$$
I've calulated the limit on Mathamatica and it equals $\sqrt{a} \sqrt{b}$.
But I've tried more to reach that value but failed .
This is what I've done:
\begin{align*} \lim _{x \rightarrow \infty}\left(\frac{a^{1 / x}+b^{1 / x}}{2}\right)^x & =\lim _{x \rightarrow \infty} e^{x \ln \left(\frac{a^{1 / x}+b^{1 / x}}{2}\right)} \\ & =\exp\left({\lim _{x \rightarrow \infty} x \ln \left(\frac{a^{1 / x}+b^{1 / x}}{2}\right)}\right) \\ & =\exp\left({\lim _{x \rightarrow \infty} x \ln \left(\frac{a^{1 / x}+b^{1 / x}}{2}\right)}\right) \\ & =\exp\left({\lim _{x \rightarrow \infty} \frac{\ln \left(\frac{a^{1 / x}+b^{1 / x}}{2}\right)}{\frac{1}{x}}}\right) \end{align*}
Also, \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\ln \left(\frac{a^{\frac{1}{x}}+b^{\frac{1}{x}}}{2}\right)}{\frac{1}{x}}\right) &=\frac{\frac{\frac{-a^{1 / x}}{2 x^2}-\frac{-b^{1 / x}}{2 x^2}}{\frac{a^{1 / x}+b^{1 / x}}{2}}}{-\frac{1}{x^2}}\\ &=\frac{-a^{1 / x}-b^{1 / x}}{a^{1 / x}+b^{1 / x}},\end{align*} so the original limit is $$\exp\left({\lim _{x \rightarrow \infty} \frac{-a^{1 / x}-b^{1 / x}}{a^{1 / x}+b^{1 / x}}}\right)$$
and stopped here!
How can I reach the value $\sqrt{a} \sqrt{b}$?
