## Thomas' Calculus 13th Edition

$\ln 2$
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. $(\ln 2) \lim\limits_{x \to \infty} (\dfrac{1}{1})=\ln 2$