Answer
$0$
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, $\lim\limits_{x \to 0} f(0)=\lim\limits_{x \to \infty} \ln (\dfrac{x}{\sin x})=\dfrac{0}{0}$
This shows an indeterminate form of the limit, so we need to use L'Hospital's rule.
Here, $A'(x)=1$ and $B'(x)=\cos x$
$ \lim\limits_{x \to 0} \ln ( \dfrac{1}{\cos x})=\ln (\dfrac{1}{\cos 0})=0$