The theory of limits that I know at the time of university in Italy has been developed in a very general way by taking as a starting point the notion of an ordered variable from an ordered set of operations.
If $[O]$ is a strictly ordered set, I remember that an ordered variable $y$ definitely assumes a certain property if it is possible to determine an operation $O^{\ast}$ such that the indicated property can be verified by the results of all the operations $O$ following $O^{\ast}$ (i.e. $O\to O^{\ast}$).
In the case of convergence I wrote, if $\epsilon\in \Bbb R, \epsilon>0$ then:
$$\ell\in\mathbb R\smallsetminus \{-\infty,+\infty \}$$ and when $O\to O^{\ast}$ $$|y-\ell|<\epsilon \iff \lim y=\ell$$
Do any of you users know the theory of ordered variables, a well written book on this theme in Italian or English, that is also used for integrals, successions, limits and what is the practical meaning?