I want a way to describe how two continuous functions $f,g \colon (X-x) \to Y$ might "share a limit" at the point $x$ when unfortunately neither of $\displaystyle \lim _{y \to x}f(x)$ or $\displaystyle \lim _{y \to x}g$ happen to exist..
Let me give an example. Suppose $x = 0$ and $f,g \colon (0,1] \to \mathbb R$ are given by $f = \sin(1/x)$ and $g = \sin(1/x) +x$. Then for any prescribed $\epsilon > 0$ there is an $\delta > 0$ such that, when I restrict the functions to the ball $B(0,\delta)$ their graphs are within distance $\epsilon$ of each other.
However in this case it's a lot easier. We can just take the difference of the functions $(f-g)(x) =x$ and say that function tends to zero at $x$. But this relies on how the real line has an additive structure. If we replace the codomain with something more exotic this definition might not apply.
So how might I formalise this notion of "agreeing at infinity" for more general topological spaces $Y$?