Question

Doar limita de la c.

Answer 1

$f:(1,+\infty)\to\mathbb{R}\,,\,\,\, f(x) = \dfrac{1}{(x-1)^2}-\dfrac{1}{x^2}\\ \\\\ \lim\limits_{n\to+\infty}\left(f(2)+f(3)+...+f(n)\right)^{n^2} =\\ \\ = \lim\limits_{n\to+\infty}\left(\dfrac{1}{1^2}-\dfrac{1}{2^2}+\dfrac{1}{2^2}-\dfrac{1}{3^2}+\dfrac{1}{3^2}-...+\dfrac{1}{(n-1)^2}-\dfrac{1}{n^2}\right)^{n^2}\\ \\ = \lim\limits_{n\to +\infty}\left(1-\dfrac{1}{n^2}\right)^{n^2}\\ \\ = \lim\limits_{n\to +\infty}\left(1-\dfrac{1}{n}\right)^{n}$

$= \lim\limits_{n\to +\infty}\left[\left(1+\dfrac{1}{(-n)}\right)^{(-n)}\right]^{-1} = e^{-1} = \boxed{\dfrac{1}{e}}\\\\\\ \underline{\text{Limite remarcabile:}}\\\lim\limits_{u(x) \to \pm\infty}\left(1+\dfrac{1}{u(x)}\right)^{u(x)} = e$