Answer
$0$
Work Step by Step
Here, we have $\lim\limits_{x \to 0} f(0)=\lim\limits_{x \to \infty} \ln (\dfrac{x}{\sin x})=\dfrac{0}{0}$
This shows an indeterminate form of limit, thus we will apply L-Hospital's rule such as:
$\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$
Now, $p'(x)=1$ and $q'(x)=\cos x$
Thus, $\lim\limits_{x \to 0} \ln ( \dfrac{1}{\cos x})=\ln (\dfrac{1}{\cos 0})$
or, $\ln (\dfrac{1}{1})=0$
