\(\displaystyle\lim_{{{n}\rightarrow\infty}}{n}{\left({x}^{{{\frac{{{1}}}{{{n}}}}}}-{1}\right)}=\lim_{{{n}\rightarrow\infty}}{\frac{{{x}^{{{\frac{{{1}}}{{{n}}}}}}-{1}}}{{{\frac{{{1}}}{{{n}}}}}}}={f}'{\left({0}\right)}\) , where \(\displaystyle{f{{\left({t}\right)}}}={x}^{{t}}\). Since

\(\displaystyle{f}'{\left({t}\right)}={\ln{{\left({x}\right)}}}{x}^{{t}}\)

it follows that \(\displaystyle{f}'{\left({0}\right)}={\ln{{\left({x}\right)}}}\)

\(\displaystyle{f}'{\left({t}\right)}={\ln{{\left({x}\right)}}}{x}^{{t}}\)

it follows that \(\displaystyle{f}'{\left({0}\right)}={\ln{{\left({x}\right)}}}\)