TITLE: Finding $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$
QUESTION [5 upvotes]: Find $\lim_{x\to +\infty}(\frac{x+\ln x}{ x-\ln x})^{\frac{x}{\ln x}}$. I tried using l'Hospital rule with the continuity of $e$ function. Also tried using Taylor expansion with no success. What should I do? Thank you.

REPLY [2 votes]: With $t=\frac{x}{\ln x}$ your limit can be written
$$
\lim_{t\to+\infty} \Bigl(\frac{1+1/t}{1-1/t}\Bigr)^t.
$$
If you know that
$$
e=\lim_{t\to+\infty}(1+1/t)^t
$$
and
$$
\frac{1}{e}=\lim_{t\to+\infty}(1-1/t)^t
$$
then you are essentially done.