$\lim_{x \to \infty} \frac {\log(|\sin(x)|)}{\log(x)}$ Does $\lim_{x \to \infty} \frac {\log(|\sin(x)|)}{\log(x)}$ exist considering $\sin(x) = 0$ When $x=n\pi$?

The limit inf is $-\infty$ and obtained when $\sin=0$, while the limit sup is $0$ and is obtained when $\sin=1$. The limit does not exist because the two limits are not equal.
