Formulas for sequences and then how to find the limsup and liminf For each of the following sequences, calculate the limit superior and the limit inferior. If the sequence converges, calculate its limit. Justify your answers.
A) $\ \ \left\{\frac{1}{2},\frac{1}{2},-\frac{1}{2},\frac{1}{3},\frac{2}{3},-\frac{2}{3},\frac{1}{4},\frac{3}{4},-\frac{3}{4},\frac{1}{5},\frac{4}{5},-\frac{4}{5},\ldots\right\}$.
B) $\ \ \left\{\frac{1}{2},1,\frac{1}{4},\frac{1}{3},\frac{1}{6},\frac{1}{5},\frac{1}{8},\frac{1}{7},\ldots\right\}$.
I can calculate the inf and sup of equations but lim inf and lim sup confuse me as well as finding the formulas for these sequences.

A) Limes superior is $1$, limes inferior is $-1$.  
The important subsequence for proving lim sup is
$$a_n =\left\{\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\ldots\right\}$$
as its limit is $1$, and for every its element are two previous elements in the original sequence smaller. 
For lim inf the reasoning is similar - with the subsequence
$$b_n =\left\{-\frac{2}{3}, -\frac{3}{4},-\frac{4}{5},-\frac{5}{6},\ldots\right\}$$
B) limit of this sequence is $0$ - it is just a "masked" sequence  
$$\left\{1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\frac{1}{8},\ldots\right\}$$
with swapped pairs of adjacent members. So lim sup and lim inf are both $0$, too.
