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, $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{\ln (x+1)}{\log_2 x}=\dfrac{\infty}{\infty}$
This shows an indeterminate form of a limit, so we need to use L'Hospital's rule.
$ \lim\limits_{x \to \infty} \dfrac{1/x+1}{1/x \ln2}=\ln 2 \lim\limits_{x \to \infty} \dfrac{x}{x+1}=\dfrac{\infty}{\infty}$
Again apply L'Hospital's rule.
$\lim\limits_{x \to \infty} f(x)=(\ln 2) \lim\limits_{x \to \infty} \dfrac{1}{1}=\ln 2$
