Surreal analysis is sort of a thing but it is quite far out there (e.g. you can have transfinite series instead of merely infinite ones). Maybe you meant nonstandard analysis (NSA), which is real analysis done with infinitesimals, but the machinery justifying it is way outside of what you'd see in even a theory-oriented intro calculus class. There was an intro calculus text (Keisler, 1976) that used infinitesimals and NSA. I don't know how it dealt with constructing them though.

https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini...