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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 $|f(x)-L|<\epsilon$.

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The limit of a function can be defined as the following value if it exists.

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 $L$.

We note that in the latter case, continuous functions automatically have the limit of a function being equal to the value of the function (assuming you learned the standard definition of continuity). Intuitivly, this says that the limit is the value we get when we stare at the function really close, or the value that the function should be.

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).

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