Find $ \lim\limits{\inf(a_{n})}$, $\lim\limits{\sup(a_{n})}$, $\inf\{a_{n}\}$, ${\sup(a_{n})}$ Given
$$a_{n}=\frac{(-1)^{n}}{n}+\frac{1+(-1)^{n}}{2}$$
Compute $$\lim\limits{\inf(a_{n})}$$
$$\lim\limits{\sup(a_{n})}$$ 
$$\inf\{a_{n}\}$$
$${\sup(a_{n})}$$ 

I will assume that you're asking about $\limsup$ and $\liminf$. (This is, in my opinion the only reasonable explanation.)
Peter suggested in his comment that you could have a look at the subsequences:
$a_{2k}=1+\frac1{2k}$
$a_{2k+1}=-\frac1{2k+1}$
Can you find limits of these subsequences? What does this tell you about $\limsup$ and $\liminf$. (I guess that you have already learned about relationship between convergent subsequences and limit inferior and limit superior. Maybe this was even in the definition of $\liminf$ and $\limsup$ - not every teacher uses exactly the same definition.)
From the above you should have some inequalities for the values of $\liminf a_n$ and $\limsup a_n$, if you are able to prove the opposite inequalities, you're done.
So for this part: Are you able to show that for each $n$ you have  $-\frac1n\le a_n \le 1+\frac1n$? What do these inequalities tell you about $\liminf a_n$ and $\limsup a_n$?
