What kind of "mathematical object" are limits?
When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly with some extra axioms thrown in here and there if needed, but the fundamental idea is that of adding additional structure on sets and relations between them. I've recently tried applying this view to calculus and have been running into some confusions. Most importantly I'm not sure how to interpret Limits. I've considered viewing them as a function that takes 3 arguments, a function, the function's domain and some value (the "approaches value") then outputs a single value. However this "limit function" view requires defining the limit function over something other then the Reals or Complexes due to the notion of certain inputs and outputs being "infinity". This makes me uncomfortable and question whether my current approach to mathematics is really as elegant as I'd thought. Is this a reasonable approach to answering the question of what limits actually "are" in a general mathematical sense? How do mathematicians tend to categorize limits with the rest of mathematics?