Solution
= Solution
{c}
The <sine> has values $1,0,-1,0$ on the four residue classes of $n$ modulo four. For $n=4k+1$, the <exponential function> gives $b_n=e^{1-1/n}\to e$; the other residue classes tend to $1,e^{-1},1$. All terms satisfy $b_n\leq e$, and the first class approaches $e$. Hence
$$
\boxed{\limsup_{n\to\infty}e^{(1-1/n)\sin(n\pi/2)}=e.}
$$