The $\varepsilon$–$\delta$ definition is: for a function $f: x \to f(x)$, the limit as $x$ tends to $a$ equals to $L$ as long as for every positve number $\varepsilon$, there exists a positive number $\delta$, such that if the distance between $x$ and $a$ (greater than 0) is less than $\delta$, then the distance between $f(x)$ and $L$ is less than $\varepsilon$.
To better illustrate what I mean, I've chosen this image below, where $x_{0}$ represents $a$. As you can see, it seems as if $L$ is a function of $a$
Epsilon-Delta Definition of a Limit. (n.d.). Brilliant. Retrieved April 23, 2021, from https://brilliant.org/wiki/epsilon-delta-definition-of-a-limit/
Best Answer
This notion of 'approaching' is only an informal way to understand. In the precise definition, it is not there. See wiki
This part is kinda sketch, it is true that $x \neq a$ because we want $0< |x-a|<\delta$;notice the strict inequality.
I'd say the way you are thinking about it is brilliant here and has made me post my own question (see here). However, as far as 'standard' basic concepts go, the limit is informally dependent on the neighbourhood of 'a' rather than of that point 'a' itself.
I wouldn't personally call it a 'function' because it is non standard, a better way to say it would be to say that the limit is dependent of neighbourhood around 'a' rather than 'a' itself.
Here it depends on your definition of $f(x)$ and it's kind of a subtle point. I had a similar doubt a long while back, have a look in this post.