Is “taking a limit” a function? Is it a procedure? A ternary operation?

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
limxk c(f(x)+g(x))=climxkf(x)+c limxkg(x),
and (my group theory is virtually non existent) appears also to be a homomorphism:
limxk(fg)(x)=limxkf(x)g(x),
etc.

Anyway, my real question is, what mathematical construct is the limit?

Answer

In general, let X,Y be topological spaces, and x0 a non-isolated point of X. Then strictly speaking, “limxx0f(x)=L” is a relation between functions f:XY and points LY (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:XY, there is at most one LY such that limxx0f(x)=L. Now, for any relation R(XY)×Y which is a partial function, we can define a corresponding function {f(XY)yY,(f,y)R}Y by sending f satisfying this condition to the unique y with (f,y)R. Then that somewhat justifies the “equality” in the notation limxx0f(x)=L, though you still need to keep in mind that it is a partial function where limxx0f(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=R, we can put a ring structure on XY by pointwise addition, pointwise multiplication, etc. Then {f:XRLR,limxx0f(x)=L} turns out to be a subring of XR, and the induced function from this subring to 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 x0 is a linear subspace of XY and the limit gives a linear transformation; if Y is a topological group, you get a subgroup of XY and a group homomorphism; and so on.)

Attribution
Source : Link , Question Author : solidstatejake , Answer Author : Daniel Schepler

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