Question

Salut, ma puteti ajuta la aceasta limita ?

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

Answer 1

$l=\lim\limits_{x\to\infty} \dfrac{1+x-xe^{\frac{1}{x}}}{1-e^{\frac{1}{x}}} \\ \\ e^{\frac{1}{x}}=t \Rightarrow \frac{1}{x} = \ln t \Rightarrow x = \dfrac{1}{\ln t} \Rightarrow t\to 1 \\ \\ l = \lim\limits_{t\to 1}\dfrac{1+\dfrac{1}{\ln t}-\dfrac{t}{\ln t}}{1-t}=\lim\limits_{t\to 1}\dfrac{1+\dfrac{1-t}{\ln t}}{1-t} = \\ \\ = \lim\limits_{t\to 1}\dfrac{\ln t - t +1}{\ln t(1-t)} \overset{L'H}{=} \lim\limits_{t\to 1}\dfrac{\frac{1}{t}-1}{\frac{1-t}{t}-\ln t} =$

$=\lim\limits_{t\to 1}\dfrac{1-t}{1-t-t\ln t} \overset{L'H}{=} \lim\limits_{t\to 1}\dfrac{-1}{-1-\ln t-1} = \dfrac{-1}{-2} = \boxed{\dfrac{1}{2}}$