Q:

$\limsup \cos(\frac{n\pi}{60})$

How do you determine the $\limsup$ / $\liminf$ of $(x_n)_n=\cos(\frac{n\pi}{60})$?
$(x_n)_n$ is bounded, so if I can find the biggest and smallest adherent points I can determine the  $\limsup$ and $\liminf$ , but I don't know how...

A:

Let $A$ be the set of cluster points of $(x_n)$. Then $A$ is bounded above by $1$ and below by $-1$. 
As $(x_{120k})$ converges to $1$, we have $1\in A$. So $\limsup x_n = \sup A = 1$.
Similarly $(x_{120k+60})$ converges to $-1$ , so $-1\in A$ and $\liminf x_n = \inf A = -1$.

