Compute $\lim_{n\to\infty}\frac{1}{n}(1+\sqrt[n]{2}+\sqrt[n]{3}+\cdots+ \sqrt[n]{n}-n)$
$\lim_{x\to\infty}\frac{x^{\ln x}}{{(\ln x)}^x}$
Compute $\lim_{n\to\infty}\frac{1}{n}(1+\sqrt[n]{2}+\sqrt[n]{3}+\cdots+ \sqrt[n]{n}-n)$
$\lim_{x\to\infty}\frac{x^{\ln x}}{{(\ln x)}^x}$
Calculation with brute force: $\lim_{x\to\infty}\frac{x^{\ln x}}{{(\ln x)}^x}=\lim_{x\to\infty}\frac{e^{\ln^2 x}}{{e^{\ln(\ln x) x}}}=\lim_{x\to\infty}e^{\ln^2 x-x\ln(\ln x)} $ Then, $\lim_{x\to\infty}\ln^2 x-x\ln(\ln x)=\lim_{x\to\infty}\frac{\frac{\ln^2 x}x-\ln(\ln x)}{\frac1x}=\frac{0-\infty}{0^+}=-\infty $ Thus, $\lim_{x\to\infty}\frac{x^{\ln x}}{{(\ln x)}^x}=\lim_{x\to\infty}e^{\ln^2 x-x\ln(\ln x)}=0 $