Answer
$\ln 2$
Work Step by Step
Consider: $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{\ln (x+1)}{\log_2 x}$
and $\lim\limits_{x \to \infty} f(\infty)=\dfrac{\infty}{\infty}$
This shows an indeterminate form of limit so we will apply L-Hospital's rule such as:
$\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$
$ \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}$
Now, again apply L-Hospital's rule.
$(\ln 2) \lim\limits_{x \to \infty} (\dfrac{1}{1})=\ln 2$
