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, $$\lim_{x\to\infty}\frac{x^{\ln x}}{{(\ln x)}^x}=\lim_{x\to\infty}e^{\ln^2 x-x\ln(\ln x)}=0 $$