The value of lim from $\underset{x\to \mathrm{\infty }}{lim} {\left\{\frac{a_1^{1/x}+a_2^{1/x}+\dots +a_n^{1/x}}{n}\right\}}^{nx},$ is

Let $x=\frac{1}{y}$. Then,

    $\underset{x\to \mathrm{\infty }}{lim} {\left\{\frac{a_1^{1/x}+a_2^{1/x}+\dots +a_n^{1/x}}{n}\right\}}^{nx}$

$\begin{array}{l}=\underset{y\to 0}{lim} {\left\{\frac{a_1^y+a_2^y+\dots .+a_n^y}{n}\right\}}^{n/y}\\ \left.=\underset{y\to 0}{lim} {\left\{1+\frac{a_1^y+a_2^y+\dots .+a_n^y-n}{n}\right\}}^{n/y}\right\}\\ =\underset{y\to 0}{lim} {\left\{1+\frac{\left(a_1^y-1\right)+\left(a_2^y-1\right)+\dots .+\left(a_{n1}^y-1\right)}{n}\right\}}^{n/y}\\ =e^{\underset{y\to 0}{lim}}\left\{\frac{a_1^y-1}{y}+\frac{a_2^y-1}{y}+\dots .+\frac{a_n^y-1}{y}\right\}\\ =e^{\log a_1+\log a_2+i\dots +\log a_n}=e^{\log \left(a_1{a}_2-a_n\right)}=a_1{a}_2{a}_3\dots a_n\end{array}$