Nonstandard analysis [0] [1] uses infinitesimals but is still completely rigorous. I haven't ever really used nonstandard analysis myself, but there's a fairly well-regarded textbook available online [2].

[0]: https://en.wikipedia.org/wiki/Nonstandard_analysis

[1]: https://math.stackexchange.com/questions/51453/is-non-standa...

[2]: https://people.math.wisc.edu/%7Ehkeisler/keislercalc-06-03-2...