Question

$\lim_{x \to \infty} \frac{1}{1-e^ \frac{1}{x} } * \frac{1}{x}$

putin ajutor va rog

Answer 1

$\lim_{x \to \infty} \dfrac{1}{1-e^{ \frac{1}{x} }} \cdot \dfrac{1}{x} \\ \\ Aplicam schimbarea de variabila: t = \dfrac{1}{x} \Rightarrow t \rightarrow \dfrac{1}{\infty} \Rightarrow t\rightarrow 0 \\ \\ \Rightarrow \lim_{t \to 0} \dfrac{1}{1-e^{ t }} \cdot t = \lim_{t \to 0} \dfrac{t}{1-e^{ t }} \overset{ (L'H.)\frac{0}{0} }=\lim_{t \to 0} \dfrac{t'}{(1-e^{ t })'} = \\ =\lim_{t \to 0} \dfrac{1}{-e^{ t }} = \dfrac{1}{-e^0} = \dfrac{1}{-1}=-1$

$\Rightarrow \boxed{\lim_{x \to \infty} \dfrac{1}{1-e^{ \frac{1}{x} }} \cdot \frac{1}{x} = -1}$