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.