Answer
$\ln 2$
Work Step by Step
L'Hospital's rule states that $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{A'(x)}{B'(x)}$
Here, $\lim\limits_{x \to \infty} f(\infty)=\lim\limits_{x \to \infty} \ln (\dfrac{2x}{x+1})=\dfrac{\infty}{\infty}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
Here, $A'(x)=2$ and $B'(x)=1$
$ \lim\limits_{x \to \infty} \dfrac{2}{1}=\ln 2$