[tex]\lim\limits_{n\to \infty}\dfrac{n}{\ln n}(\sqrt[n]{n}-1) =\lim\limits_{n\to \infty}\dfrac{1}{\frac{1}{n}\ln n}(\sqrt[n]{n}-1)=\lim\limits_{n\to \infty}\dfrac{1}{\ln \sqrt[n]{n}}}(\sqrt[n]{n}-1)\\ \\ \sqrt[n]{n} = t\Rightarrow t\to 1 \\l =\lim\limits_{n\to \infty}n^{\frac{1}{n}} \Leftrightarrow \ln l = \lim\limits_{n\to \infty} \dfrac{\ln n}{n}= \lim\limits_{n\to\infty}\dfrac{\dfrac{1}{n}}{1} = 0 \Leftrightarrow l = e^0 = 1[/tex]
[tex]\lim\limits_{n\to \infty}\dfrac{1}{\ln \sqrt[n]{n}}}(\sqrt[n]{n}-1) =\lim\limits_{t\to 1}\dfrac{t-1}{\ln t} = \lim\limits_{t\to 1}\dfrac{1}{\dfrac{1}{t}} = \boxed{1}[/tex]
