Does $\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right)$ exist? I have this limit:
$$\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right)$$
I don't even know if it exists. If so, what its value ?
Really don't have any idea..

Hint: When $n$ is even, we have:
$$\lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1}\right) = \lim_{n\rightarrow\infty}\sin\left(\pi\sqrt[3]{n^{3}+1} - \pi n\right) = \lim_{n\rightarrow\infty}\sin\left(\frac{\pi}{...}\right)$$
Similarly, we can deal with the odd $n$ case and conclude the limit is $0$.
