Answer
$$0$$
Work Step by Step
Definition of a sequence defined by a function: Let us consider a function $f(x)$ whose limit $\lim\limits_{x \to \infty} f(x)$ exists then the sequence $a_n=f(n)$ will converge to the same limit.
That is, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} f(x)$
Here, in this problem we have $a_n=\ln [\sin (1/n)]+\ln n$ and $f(x)=\ln [\sin (1/x)]+\ln (x)$
Next, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} \ln [\sin (1/x)]+\ln (x)$
or, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} \ln (\dfrac{\sin (1/x)}{1/x})$
Suppose that $\dfrac{1}{x}=a$ when $t \to 0$, then $x \to \infty$
$\lim\limits_{n \to \infty} a_n=\ln ( \lim\limits_{a \to 0} \dfrac{\sin (a)}{a})$
Thus. $\lim\limits_{n \to \infty} a_n=\ln (1)=0$
