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

limx→k c(f(x)+g(x))=climx→kf(x)+c limx→kg(x),

and (my group theory is virtually non existent) appears also to be a homomorphism:

limx→k(fg)(x)=limx→kf(x)g(x),

etc.Anyway, my real question is, what mathematical construct

isthe limit?

**Answer**

In general, let X,Y be topological spaces, and x0 a non-isolated point of X. Then strictly speaking, “limx→x0f(x)=L” is a *relation* between functions f:X→Y and points L∈Y (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:X→Y, there is *at most one* L∈Y such that limx→x0f(x)=L. Now, for any relation R⊆(X→Y)×Y which is a partial function, we can define a corresponding function {f∈(X→Y)∣∃y∈Y,(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 limx→x0f(x)=L, though you still need to keep in mind that it is a *partial* function where limx→x0f(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 X→Y by pointwise addition, pointwise multiplication, etc. Then {f:X→R∣∃L∈R,limx→x0f(x)=L} turns out to be a subring of X→R, 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 X→Y and the limit gives a linear transformation; if Y is a topological group, you get a subgroup of X→Y and a group homomorphism; and so on.)

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