[zozbot234]

> The rigorous form of the non-rigorous version is non-standard analysis

This is quite overstated. There are other approaches to infinitesimals such as synthetic differential geometry (SDG aka. smooth infinitesimal analysis) that are probably more intuitive in some ways and less so in others. SDG infinitesimals lose the ordering of hyperreals in non-standard analysis and force you to use some non-classical logic (intuitively, smooth infinitesimals are "neither equal nor non-equal to 0", wherein classical reasoning would conflate every infinitesimal with 0), but in return you gain nilpotency (d^n = 0 for any infinitesimal d) which is often regarded as a desirable feature in informal reasoning.