Question

Cât este lim x-> infinit din radical de ordin x din x

Answer 1

$l =\lim\limits_{x\to \infty} \sqrt[x]{x} = \lim\limits_{x\to \infty}x^{\frac{1}{x}} \\ \\\Rightarrow \ln l = \ln\left(\lim\limits_{x\to \infty}x^{\frac{1}{x}}\right)\\ \\ \Rightarrow\ln l = \lim\limits_{x\to \infty} \ln (x^{\frac{1}{x}})\\ \\\Rightarrow \ln l = \lim\limits_{x\to \infty} \frac{1}{x}\cdot \ln x\\\\\Rightarrow\ln l = \lim\limits_{x\to \infty} \dfrac{\ln x}{x}\\ \\\Rightarrow \ln l \overset{^{\frac{\infty}{\infty}}}{=} \lim\limits_{x\to \infty} \dfrac{\frac{1}{x}}{1}\\ \\ \Rightarrow\ln l = 0\\ \\ \Rightarrow \boxed{l = 1}$