Question

Alguém consegue me ajudar nessa questão .

Answer 1

$\displaystyle L=\lim\limits_{n\to +\infty}\left(\frac{n+1}{n+3}\right)^{n+1}\\ \\ L=\lim\limits_{n\to +\infty}\left(1-\frac{2}{n+3}\right)^{n+1}\\ \\ L=\lim\limits_{n\to +\infty}\left(1+\dfrac{1}{-\dfrac{n+3}{2}}\right)^{-\dfrac{n+3}{2}\cdot \dfrac{2(n+1)}{n+3}}\\ \\ L=\lim\limits_{m\to -\infty}\left(1+\dfrac{1}{m}\right)^{m\cdot \lim\limits_{n\to+\infty}\dfrac{2(n+1)}{n+3}}$

$L=(1/e)^{\lim\limits_{n\to+\infty}\dfrac{2(n+1)}{n+3}}\\ \\ L=(1/e)^{\lim\limits_{n\to+\infty}\dfrac{2(1+\dfrac{1}{n})}{1+\dfrac{3}{n}}}\\ \\ L=(1/e)^{\dfrac{2(1+0)}{1+0}}\\ \\ \boxed{L=e^{-2}}$