Consider the sequence $\{(n,n)\}_{n\in\mathbb{N}}.$ $(n,n)\to(\infty,\infty)$ as $n\to\infty$, but "$(\infty,\infty)$'' loses the information that the terms of $(n,n)$ remain equal $\forall n$. This is contrast to the sequence $\{(n,n^n)\}_{n\in\mathbb{N}}, $ where $(n,n^n)\to(\infty,\infty)$ but $\lim\limits_{n\to\infty}\frac{n}{n^n}=0$.
Question: I want to parsimoniously express $\lim\limits_{n\to\infty} (n,n)$ in a way that preserves the information that both terms of this double stay equal as $n\to\infty$. Is $``\!\lim\limits_{n\to\infty} (n,n)\!"$ the most succinct way this can be done?
Losing the information that two elements are the same, or more generally are not independent, is indeed a common problem. E.g. when taking into account computation error, or observation error, adding the error margins is valid only when variables are independent. $x-x$ has no error at all.
This problem occurs every time an element is replaced by a class of equivalence - here you mentioned equivalence at $+\infty$. For example the pointer aliasing problem in computer science is another - quite remote - example.
The most precise thing one can do is to keep elements in a symbolic form (in a form that uniquely identifies them) as long as possible. I.e. keeping $n,n$ this way, or $f(n), f(n)$.