The value of $\underset{\mathrm{x}\to \mathrm{\infty }}{lim} {\left[\frac{1^{1/\mathrm{x}}+2^{1/\mathrm{x}}+3^{1/\mathrm{x}}+\dots +{\mathrm{n}}^{1/\mathrm{x}}}{\mathrm{n}}\right]}^{\mathrm{nx}}$ is

$\begin{array}{l}\underset{x\to \mathrm{\infty }}{lim} {\left(\frac{1^{1/x}+2^{1/x}+3^{1/x}+\dots +n^{1/x}}{n}\right)}^{nx}\\ =\underset{y\to 0}{lim} {\left(\frac{1^y+2^y+3^y+\dots +n^y}{n}\right)}^{\frac{n}{y}}\\ =e^{\underset{y\to 0}{lim} \frac{n}{y}\left(\frac{1^y+2^y+3^y+\dots +n^y}{n}-1\right)}\\ =e^{\underset{y\to 0}{lim} \left(\frac{1^y+2^y+3^y+\dots +n^y-n}{y}\right)}\\ =e^{\underset{y\to 0}{lim} \left[\frac{\left(1^y-1\right)}{y}+\frac{\left(2^y-1\right)}{y}+\frac{\left(3^y-1\right)}{y}+\dots +\frac{\left(n^y-1\right)}{y}\right]}\\ =e^{(\log 1+\log 2+\log 3+\dots +\log n)}\\ =e^{\log (1\cdot 2\cdot 3\dots n)}=n!\end{array}$