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?



Terminology for sequences/functions that approach each other
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".


