Different definitions of a limit of a function in $\mathbb R$. Are they equivalent or not? If a limit $L$ of a function $f:A\to\mathbb R$ exists at a point $a\in \mathbb R$, where $A\subset\mathbb R$ is a proper subset of the set of real numbers, is there any difference between the statements "the limit exists when $a$ is an accumulation point" and "the limit exists when $f$ is defined on a deleted neighbourhood of $a$"?

My motivation for asking this question is because of an answer to a previous question of mine.
You cannot approaches to any $a \in Dom(f)$. $a$ must be a limit point but this  not implies that the limit exists. e.g., $f(x)=\frac{1}{x}$, $x \in [-1,1]=A$,  $x=0$  is a limit point for $A$ but as $x\to 0$, $f\to \pm\infty$.
You cannot approaches to any $a \in Dom(f)$. $a$ must be a limit point but this  not implies that the limit exists. e.g., $f(x)=\frac{1}{x}$, $x \in [-1,1]=A$,  $x=0$  is a limit point for $A$ but as $x\to 0$, $f\to \pm\infty$.