TITLE: Terminology for sequences/functions that approach each other
QUESTION [6 upvotes]: 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?

REPLY [4 votes]: You could say $ a $ and $ b $ are mutually asymptotic. Example: "Since $ n -1/n$ and $n+1/n $ are mutually asymptotic. .." And usually even drop the word "mutually".