From what I read on the subject, this _was_ the main way of dealing with calculus, but it was not rigorous. The first rigorous definition of a limit was given using the epsilon-delta approach, by Weierstrass, with of course many many contributions by his predecessors.

Everybody switched during the 19th century, because epsilon-delta was in fact the only rigorous method.

Abraham Robinson gave a rigorous foundation for infinitesimals only in 1960.