Many years ago, when I was at the university, I learned all the theory of limits of a real function of real variable by means of ordered variables. Precisely I was taught to consider a set $\mathrm{O}$ of operations, strictly order this set by using a precise ordering binary relation called $\color {red}{\preccurlyeq}$ and get $[\mathrm{O}]$, the set of ordered (or well ordered) variables. I think this is a prefilter ordering, but I admit I haven't fully understood it, even after about $30$ years.
But what exactly this ordering does? Are the operations we are talking about all the operations that we use to in order to calculate a limit or operations that provide the definition of a limit?
Chapter II of the text [1] by M. Picone and G. Fichera (but this path is also followed by Carlo Miranda in the university text [2]) is devoted to the foundations of infinitesimal analysis by using this method; it is perhaps in this book that the didactic address of Picone is manifested in the most radical way. The theory of limits is carried out in a very general form, by considering the notion of an ordered variable belonging to an ordered set of ordered operations, and proving as the first application of the use of these concepts the existence minimum and maximum limits, i.e. $\liminf$ and $\limsup$.
Despite all of my efforts, I've never understood why thus approach is useful, although this way of doing things impressed me as something stuck in my mind.
If anyone could elaborate and explain this approach to me I would be grateful.
Furthermore, if someone needs an English translation of the few relevant pages on this theme from my book in Italian I am available for the task.
Sources
[1] Mauro Picone, Gaetano Fichera, Trattato di analisi matematica. Vol. I: Fondamenti dell'algebra e del calcolo infinitesimale nel campo reale e in quello complesso (in Italian), Tumminelli Editore: Roma, pp. vi+520 (1954), MR0106814, Zbl 0058.03803 (here is link a link to a review by Carlo Miranda).
[2] Carlo Miranda, Lezioni di analisi matematica. Volume I, Liguori Editore: Napoli, pp. 528 (1993), ISBN: 978-88-20704445.