I found this definition on wikipedia.
A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself.
But doesn't this just mean it could be pretty much any point? Not necessarily anywhere near a boundary/limit of S? Just a random point in S.
For example pick the point, 2, in S (the standard topology of R between (1,3)). Every neighbourhood of x will contain a point in S that is not x.
What am I missing, this notion of a limit point seems pointless..?