Calculate the limit
$$ \lim_{n\rightarrow\infty}(\sin(n!)+1)^{1/n} $$
or prove that the limit does not exist.
This appeared as a problem in my mathematical analysis test, and the answer was that the limit exists and it was $1$. But later the teacher found a mistake in his proof and eventually removed the problem from the test. But I'm just curious. Does this problem have a certain answer?
The biggest question for me, is that I can't show that there does not exist any $n_0$ so that $\sin(n_0!)$ is close to $-1$ enough so that the original term might not converge. Any help would be appreciated!