is the limit point of $(-1)^n$ in contradiction to the definition of a limit point? $(-1)^n=\{-1,1,-1,1,...\}$
Now according to the definition of limit point
A point $x$ in $X$ is a limit point or cluster point or accumulation point of a set of $S$ if every neighborhood of $x$ contains at least one point of $S$ different from $x$ itself
Now if I take deleted nbd of $1$ I don't have any element in the set. So how $1$ or $-1$ would be its limit point.

Just as jjagmath says in his comment,  the concept of a limit point is different for sequences and sets.
If you were to consider $\{-1,1,-1,...\}$ as a set, then it would actually be $\{-1,1\}$ since sets do not contain the same element more than once (unless it is a multiset). In this case you are correct, this set does not have limit points.
However, if you consider the sequence $-1,1,-1,...$, then a limit point is one that for every neighbourhood around the point, there are infinitely many terms that lie within the neighbourhood. So $-1$ and $1$ are limit points of the sequence because they occur infinitely often.
