Can limits be thought of as linear functionals (or operators, depending on context)?

Ok so I just started Calc I this summer and since I already feel pretty comfortable with it from high school, I'm trying to gain a more rigorous perspective on it. I already know that limits behave linearly in the sense that $$ \lim_{x \to a}[f(x)+g(x)]=\lim_{x \to a}f(x)+\lim_{x \to a}g(x) $$ and $$ \lim_{x \to a}[af(x)]=a \left(\lim_{x \to a}f(x)\right) $$ but I have never seen them formally described as a linear functional (or linear operator if the output is a function as in the case of the derivative) in the sense that they take an element of a suitable function space (for simplicity, take the continuous functions which form an infinite dimensional normed vector space, lets say $E$) such that $L:E \to \mathbb{R}$ where $L$ is defined by $$ L=\lim_{x \to a} $$ My gut instinct on this is that it may have never been useful to formalize the notion of a limit as a linear functional or that the definition of the derivative operator $D:C^{k} \to C^{k-1}$ as $$ Df=\lim_{h \to 0} \frac{f(x+h)+f(x)}{h} $$ makes this so obvious that no one talks about it explicitly. Another way to phrase my question would be:

"Limits belong to which class of mathematical objects?"

I tried asking my teacher but she didn't even understand what I was asking (she is a TA type who is well intentioned but clearly not comfortable enough with the material to teach) so any additional insights would be of great help here.

Limits can definitely be seen as functionals. The problem is that to consider them as a functional, you need the limit to be defined on all elements of a vector space.

If you are dealing with continuous functions on a compact set (or an interval, to make things simpler), then the limit is just evaluation at a point. As soon as you are dealing with non-continuous functions (a very common occurrence in functional analysis, as in $L^p$ spaces for instance), limits as-is make no sense.

Still, one can use deep ideas in functional analysis to extend a limit from some objects where it exists to a more general setting. For instance, you could consider $\ell^\infty(\mathbb N)$, the set of bounded sequences. Of course, not every sequence has a limit; but there is a way to define a linear functional (many, actually) that agree with the limit where it exists. Some keywords for these ideas (I'll let you read about them if you have the background) are