Limits work just fine without dealing with infinity. When you do an epsilon delta proof, you only ever deal in the finite. You can also have continuity while on

Limits work just fine without dealing with infinity. When you do an epsilon delta proof, you only ever deal in the finite. You can also have continuity while only dealing in finite constructions. The real numbers are a very convenient abstraction, but they are also a leaky abstraction (edit: when paired with axiom of choice), which is anathema to mathematics. We can still study the real numbers, they are still valid in some abstract sense, and they are convenient, but an element of the set of reals minus computables can never be constructed, so you'll never actually need to deal with them.