I like the standard "epsilon/delta" approach better. In that approach the idea of "infinite" insofar as it applies to real numbers is introduced solely as a not

I like the standard "epsilon/delta" approach better. In that approach the idea of "infinite" insofar as it applies to real numbers is introduced solely as a notational convenience, and is in no way necessary. All notions of limits can exist without infinity, and in this way I believe they are much more logically clear.