Sequence: $\displaystyle \frac{1}{2},\frac{1}{4},\frac{1}{2},\frac{3}{4},\f rac{1}{8},\frac{1}{4},\frac{3}{8},\frac{1}{2},\fra c{5}{8},\frac{3}{4},\frac{7}{8},.......$
Find all limit points
We can't really find out the limit points if we don't know how to define the sequence. The blue sequence above diverges; it's just an arithmetic sequence: $\displaystyle a_n=a_{n-1}+\frac{1}{8}$.
I think we would need more information before we could say definitively what this sequence is doing.