Definition of the limit of a function at a point

What is the formal definition of the limit of a function at a point in Real Analysis?

Is it this one?

For a function $f: D\subseteq \mathbb{R}\rightarrow \mathbb{R}$ with the domain $D$ containing an open interval around $a$, except possibly at $x=a$, say that $\lim_{x\rightarrow a}f(x)=L$ if $\forall \epsilon>0$, there exists $\delta>0$ so that for any $x$ with $0<|x-a|<\delta$, we have $|f(x)-L|<\epsilon$.

Given a sequence of points, $\{a_n\}$ going to $a$ (say $a$ is the value of the function), we can, for any $\epsilon>0$, find $n$ such that $|L-f(a_m)|<\epsilon$ for all $m>n$. We say that limit is $L$
It can also be defined as follows: Given $\epsilon>0$ when we can find $\delta>0$ such that for all $x, y$ such that $|x-a|<\delta$, we have $|f(x)-L|<\epsilon$. We say the limit is $L$.
EDIT: Yes, your definition is the same, note that I didn't specify the domain in either definition, so it is simply the second, forgetting the point $f(a)$ if $f(a)$ is defined. This definition is slightly more general, allowing us to define the limit of the function on the domain $\{ \frac{1}{n}|n\in \mathbb{N}\}$ and the like, and also in more general spaces (like matrices and the determinate).