Little oh notation question stuck on the limit

Question

I want to prove that $$\ln(\sin(x))=o(\ln^2(x)) (x\rightarrow\infty) $$ And i start with the definition of little oh. $$\lim_{x\to\infty}\left(\frac{\ln(\sin(x))}{\ln^2(x)}\right)=0$$ I don't know how find that the limit above is equal to zero as x goes to infinty.

Answer 1

If you pick the sequence of $x$ where $\sin x = 0$, you see the limit can't possibly exist. Also, remember $\sin x$ takes on negative values, and there the logarithm isn't even defined.