Question

Salut, aveti o idee de rezolvare la problema 249...?

Answer 1

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

$l = \lim\limits_{t\to 1}\dfrac{t^{t^{t^t^{...}}}}{\ln t^{t^{t^t^{...}}}}(t-1) = \lim\limits_{t\to 1}\dfrac{ t^{t^{t^t^{...}}}}{{t^{t^t^{...}}}\ln t}(t-1) = \lim\limits_{t\to 1}\dfrac{t-1}{\ln t} = \\ \\ = \lim\limits_{t\to 1}\dfrac{1}{\dfrac{1}{t}} = \boxed{1}$