Let $(a\,;b)$ be an open interval of real numbers.
Let $L$ be some real number.
Iff ${\forall}\epsilon \in \mathbb R_{>0}:\exists\delta \in \mathbb R_{>0}:\forall x \in (a\,;b):0<|c-x|<\delta\implies |f(x)-L|<\epsilon$ then we say that $L=\displaystyle \lim_{x\rightarrow c}f(x)$.
So far I know that $c$ is some constant real number. I've also heard (not from every source though) that it needs to be a limit point of $(a\,;b)$. Is this a tautology, or are there cases where the proposition defining $L$ above is true, but $c$ isn't a limit point, therefore (?) making $L$ not a limit in those cases?
Is saying that $c$ represents a constant real number enough, or is this definition of $c$ incomplete?