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