Confusing definition of limit points 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..?

Intuitively, when we say that some $x \in X$ is a limit point of $S$, we mean that that there are infinitely many elements of $S$ that are densely packed near $x$. Indeed, we can get arbitrarily close to $x$ using only elements of $S$.
For example, you may have heard that $\mathbb Q$ is dense in $\mathbb R$. A consequence of this fact is that any irrational number, such as $x = \pi$, is a limit point of $S = \mathbb Q$ and thus can be approximated to an arbitrary precision using only rational numbers such as:
$$
3, 3.1, 3.14, 3.141, 3.1415, 3.14159, \ldots
$$
