What do I call two sequences $a, b$ such that $\lim_{n\to\infty} |a_n - b_n| = 0$? Or what do I call two functions $f, g$ such that $\lim_{x\to c} |f(x) - g(x)| = 0$? (For my purposes, these are essentially the same thing, and I will happily extend a term for one to the other.)

This seems like such a straightforward condition that it must have a standard term, but I can't find it (in either context). I looked through the Wikipedia article on all of the variations of big-$O$ and the like, but these are all too weak. If $\lim_n a_n$ (hence $\lim_n b_n$) existed, then I could call $a$ and $b$ ‘coterminal’, but that limit might not exist. In an incomplete space, I have seen $a$ and $b$ called ‘co-Cauchy’ under the weaker assumption that one (hence both) is Cauchy, but I don't want to assume that either. I could call $\exp f$ and $\exp g$ ‘asymptotic’ (as $x \to c$), but I want to refer to $f$ and $g$ directly.

Surely somebody knows a term for this?

shouldmean what I want, but when I look it up, it doesn't. – Toby Bartels Jul 1 '13 at 0:27