Question

por favor resolver el sgte.limite:​

Answer 1

$L=\lim\limits_{n\to\infty}\left(\dfrac{a^{1/n}+b^{1/n}}{2}\right)^n\\\\\\L=\lim\limits_{n\to\infty}\exp\left\{n\ln\left(\dfrac{a^{1/n}+b^{1/n}}{2}\right)\right\}\\\\\\L=\lim\limits_{n\to\infty}\exp\left\{\dfrac{\ln\left(\dfrac{a^{1/n}+b^{1/n}}{2}\right)}{\dfrac{1}{n}}\right\}$

$L=\exp\lim\limits_{n\to\infty}\dfrac{\ln\left(\dfrac{a^{1/n}+b^{1/n}}{2}\right)}{\dfrac{1}{n}}\right\\ \\\\\text{Indeterminaci\'on }0/0,\text{Apliquemos L'Hospital}\\\\\\L=\exp\lim\limits_{n\to\infty}\dfrac{-\dfrac{1}{2n^2}(a^{1/n}\ln a+b^{1/n}\ln b)}{-\dfrac{1}{n^2}}\\\\\\L=\exp\lim\limits_{n\to\infty}\dfrac{a^{1/n}\ln a+b^{1/n}\ln b}{2}\\\\\\L=\exp\left(\dfrac{\ln a+\ln b}{2}\right)\\\\\\L=\exp\left(\dfrac{\ln (ab)}{2}\right)\\\\\\L=\exp\ln\sqrt{ab}\\\\\boxed{\boxed{L=\sqrt{ab}}}$