Q:

What is $\lim_{n \rightarrow \infty} \frac{n ^ {(1 + \sin{(n)})}} {\sqrt{n}}$?

What is the value of this limit : 
$$\lim_{n \rightarrow \infty} \frac{n ^ {(1 + \sin{(n)})}} {\sqrt{n}}$$
The original question was to find out the asymptotic relation between $n^{(1 + \sin{(n)})}$ and $\sqrt{n}$.

A:

For infinitely many $n$, $\sin n < -3/4$,  so along this subsequence, $$\frac{n^{1+\sin n}}{\sqrt{n}} ≤ n^{-1/4} → 0.$$
However, for infinitely many other $n$, $\sin n >  0$, so here we see
$$ \frac{n^{1+\sin n}}{\sqrt n} ≥ n^{1/2} → ∞. $$

