Value of limsup i? This is a part of my question.
$\lim \sup \cos(n\pi/12)$ as n goes to infinity
What is the value of this limit? 

When a sequence is bounded, the limsup is the largest limit of all convergent subsequences.
For all $n$, $\cos(n\pi/2)\leq 1$ so $\limsup\cos(n\pi/2) \leq 1$.
And for the extraction $n=4k$, $\cos(n\pi/2)=\cos(2k\pi)=1$. So $\limsup\cos(n\pi/2)\geq 1$.
Hence
$$
\limsup_{n\rightarrow +\infty}\cos(n\pi/2)=1.
$$
