prove: $\displaystyle \lim_{n}\cfrac {ln|sin(n)|} {n} = 0 , n\rightarrow \infty$
or show it is not true.
thanks
I think the limit does not exist but i`m not sure.
Try with L`hopital rule , you will get limn->inf from cot(n) , and that does not exist as the function is oscilating to infinity without aproaching any specific value.
Sry for bad english
Since $\displaystyle \begin{align*} \sin{(x)} \end{align*}$ does not have an infinite limit, then I don't see how $\displaystyle \begin{align*} \ln{ \left| \sin{(x)} \right| } \end{align*}$ could possibly have one. So I would expect that the limit does not exist. Thus L'Hospital's Rule should not be used...
I would try to show that for any fixed $\displaystyle \varepsilon>0$ there are infinitely many n's such that $\displaystyle \sin x>1-\varepsilon$ and infinitely many n's such that $\displaystyle |\sin x|<\varepsilon$.
The basic idea is to use Dirichlet principle.
Similar approach was used here: Sine function dense in [−1,1] at MSE.