I want to introduce function symbols $\lim_{x\to a}$ in the first-order language.
First, we introduce limit as a function symbol $fun$ for the sake of simplicity. $\forall f, a\exists! b\psi(b, f, a)$ is true as $\psi(b, f, a) = [f$ is real function] $\land\forall\epsilon>0\exists\delta>0\forall x\in\mathbb{R}$ [$0 < | x − a | < \delta \to | f(x) − b | < \epsilon
]$, so by "extension by definition" we can have introduced the function symbol $fun$ in the language.
However $fun$ is one, but $\lim_{x\to a}$ is multiple and $\lim_{x\to a}$ have "argument $a$".
How can the symbols $\lim_{x\to a}$ be introduced?
First, let $\varphi(a, f, b)$ be a formula saying that $a$ is a real, $f$ is a real function and $b$ is a real that is a limit of $f$ at $a$:
$$\begin{align} \varphi(a, f, b) \triangleq \ & f \colon \mathbb{R}\to\mathbb{R} \ \land a\in\mathbb{R} \land b\in\mathbb{R} \ \land \\ &(\forall\epsilon>0)(\exists\delta>0)(\forall x\in\mathbb{R})\ [0 < | x − a | < \delta \Rightarrow | f(x) − b | < \epsilon ]. \end{align} $$ Here we assume that $u > 0$ implies that $u\in\mathbb{R}$.
Now let $\psi(a, f, y)$ say that either $\varphi(a, f, b)$ holds, or $b = \emptyset$ and there is no $y$ for which $\varphi(a, f, y)$ holds:
$$\begin{align} \psi(a, f, b) \triangleq & \ \varphi(a, f, b) \lor [\neg\ (\exists y)\ \varphi(a, f, y) \land b = \emptyset] \end{align}$$ The second clause handles the cases where one or more variables is of the wrong type, or no limit exists.
I'll assume we're working in ZFC, though ZF + DC (Dependent Choice) would suffice. Consider the following set comprehension:
$$ L = \{(a, F)\mid a\in\mathbb{R} \land F \colon \mathbb{R}^{\mathbb{R}}\to \mathbb{R} \land (\forall f\in \mathbb{R}^{\mathbb{R}})\ \psi(a, f, F(f)) \} \tag{*} $$
It's a theorem of ZFC that $L$ exists (is a set), $L$ is a function, and that for all $a\in\mathbb{R}$ there is a unique F such that $(a, F)\in L$.
Finally, let $\theta(a, F)$ say that either $(a, F)\in L$, or $a\notin \mathbb{R}$ and $F = \emptyset$. Then it's a theorem of ZFC that for all $a\in\mathbb{R}$ there's a unique $F$ such that $\theta(a, F)$: $$ \operatorname{ZFC}\vdash (\forall a)(\exists! F)\ \theta(a, F) $$ This entitles us to introduce $\operatorname{lim}$ by definition, with the defining axiom: $$ (\forall a)\ \theta(a, \operatorname{lim}(a)) $$ If $a\notin\mathbb{R}$, then $\operatorname{lim}(a) = \emptyset$, and if $a\in\mathbb{R}$, then for any real function $f$,
$$ \operatorname{lim}(a)(f) = \begin{cases} \text{limit of } f \text{ at }a, & \text{if the limit exists;} \\ \emptyset, & \text{otherwise}. \end{cases} $$
You're free to write "$\operatorname{lim}_{x\to a}$" for $\operatorname{lim}(a)$, and by abuse of notation "$\operatorname{lim}_{x\to a}f(x)$" instead of $\operatorname{lim}_{x\to a}(f)$. The dummy variable $x$ serves no great purpose in this example, but it's clearly helpful with functions of more than one variable — for example, $\operatorname{lim}_{u\to a}g(u, v)$.