Is "taking a limit" a function? Is it a procedure? A ternary operation?

In general, let $X, Y$ be topological spaces, and $x_0$ a non-isolated point of $X$. Then strictly speaking, "$\lim_{x\to x_0} f(x) = L$" is a relation between functions $f : X \to Y$ and points $L \in Y$ (the equality notation being misleading in general).

Now, if $Y$ is a Hausdorff topological space, it happens that this relation is what is known as a partial function: for any $f : X \to Y$, there is at most one $L \in Y$ such that $\lim_{x\to x_0} f(x) = L$. Now, for any relation $R \subseteq (X \to Y) \times Y$ which is a partial function, we can define a corresponding function $\{ f \in (X \to Y) \mid \exists y \in Y, (f, y) \in R \} \to Y$ by sending $f$ satisfying this condition to the unique $y$ with $(f, y) \in R$. Then that somewhat justifies the "equality" in the notation $\lim_{x\to x_0} f(x) = L$, though you still need to keep in mind that it is a partial function where $\lim_{x\to x_0} f(x)$ is not defined for all $f$. (This part relates to the answer by José Carlos Santos.)

Building on top of this, in the special case of $Y = \mathbb{R}$, we can put a ring structure on $X \to Y$ by pointwise addition, pointwise multiplication, etc. Then $\{ f : X \to \mathbb{R} \mid \exists L \in \mathbb{R}, \lim_{x\to x_0} f(x) = L \}$ turns out to be a subring of $X \to \mathbb{R}$, and the induced function from this subring to $\mathbb{R}$ is a ring homomorphism. (More generally, this will work if $Y$ is a topological ring. Similarly, if $Y$ is a topological vector space, then the set of $f$ with a limit at $x_0$ is a linear subspace of $X \to Y$ and the limit gives a linear transformation; if $Y$ is a topological group, you get a subgroup of $X \to Y$ and a group homomorphism; and so on.)

Let $I\subset\mathbb R$ be a subset of $\mathbb R$ such that $k$ is an accumulation point of $I$. Let$$R=\left\{f\colon I\longrightarrow\mathbb R\,\middle|\,\lim_{x\to k}f(x)\text{ exists}\right\}.$$Then $(R,+,\times)$ is a ring and the map$$\begin{array}{ccc}R&\longrightarrow&\mathbb R\\f&\mapsto&\lim_{x\to k}f(x)\end{array}$$is a ring homomorphism.

To be more precise, "taking $\limsup$ of a sequence" and "taking $\liminf$ of a sequence" are both functionals, i.e. linear functions from a vector space whose elements are sequences to its underlying field of real numbers. (You can also consider it a functional in a more advanced way, by putting a topology on the vector space for which the operation is continuous.)

This captures just the additivity. To capture multiplicativity, you need to consider the space of sequences not just as a vector space, but as a ring (as before, you can put a topology on this ring, in which case it becomes natural to consider a space of sequences satisfying enough conditions to become a Banach algebra). Then $\limsup$ and $\liminf$ become ring homomorphisms from a ring whose elements are sequences to the real numbers.