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. 
 A: 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

*

*Banach limits

*Free ultrafilters

*Stone-Cech compactification.

