Let X be a metric space, E a subset of X, and x a boundary point of E.
It is clear that if x is not in E, it is a limit point of E.
Similarly, if x is in E, it is a limit point of X\E.
And there are ample examples where x is a limit point of E and X\E.
My question is: is x always a limit point of both E and X\E? I have looked through similar questions, but haven't found an answer to this for a general metric space. A counterexample would be appreciated (if one exists!)
$A=\{0\}$ (in the reals, usual topology) has $0$ in the boundary, as every neighbourhood of it contains both a point of $A$ (namely $0$ itself) and points not in $A$. But it is not a limit point of $A$ as neighbourhoods of it do not contain other points from $A$ that are unequal to $0$. You need isolated points for such examples.
If you mean limit point as "every neighbourhood of it intersects $A$", boundary points are limit points of both $A$ and its complement. But I gathered from your remarks that points in the boundary of $A$ but not in $A$ are automatically limit points that you probably mean the stricter definition that I used above.