Do limsup and liminf exist only for oscillating sequences? In many sites including wikipedia, limsup and liminf are defined using the pictures of oscillating sequences.

So, is there only this type of sequence which can have limsup and liminf?
(Ok, this is reasonable that a sequence jumping up & down can have limsup and liminf,but . . .)
 A: A sequence has a finite limsup and liminf if and only if it is bounded.
On the other hand if it is unbounded upwards, its limsup will be $\infty$. (And similarly for downwards unbounded sequences).
A: $\limsup$ and $\liminf$ are generally first defined for bounded sequences so that the limit superior and the limit inferior become real numbers. 
The picture above found in Wikipedia does not seem to depict a converging sequence. But it is bounded. The picture is used to highlight the fact that although the sequence oscillates and does not converge $\limsup$ and $\liminf$ still exist.
But when you define $\limsup$ and $\liminf$  for unbounded sequences as well by including symbols $\infty$ and $- \infty$ we come to the very useful fact that every sequence has both. So in that sense, No. This is not the only type of sequence that has a limit superior and a limit inferior. In fact, in a general sense any sequence does. 
A: If a sequence converges, its limit superior is equal to its  limit inferior and it is equal to its limit.
