I was sitting in analysis yesterday and, naturally, we took the limit of some expression. It occurred to me that "taking the limit" of some expression abides the rules of a linear transformation $$\lim_{x \rightarrow k}\ c(f(x)+g(x)) = c \lim_{x \rightarrow k} f(x) + c\ \lim_{x \rightarrow k} g(x),$$ and (my group theory is virtually non existent) appears also to be a homomorphism: $$\lim_{x \rightarrow k} (fg)(x) = \lim_{x \rightarrow k} f(x)g(x), $$ etc.
Anyway, my real question is, what mathematical construct is the limit?