I'm of the opinion that there's a reason why a subset (myself included) of people who when initially exposed to infinitesimals, and specifically the part where you start just disregarding terms, reject them (it's one of the oldest arguments related to calculus! [0]). Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor hurts those arguments until you have a rigorous justification for them (that didn't appear until the 1960's if my memory is right).
I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.)
> Those geometric arguments are essentially less rigorous versions of limits! And that lack of rigor (...)
Yes it's equivalent to limits, but limits are a very cumbersome machinery, specially if you use the epsilon delta definition (there exists .. such that all ..).
But note that I just linked you a PDF that does fully 100% rigorous calculus using only infinitesimals with no limits. Yhey aren't disregarding small terms willy nilly (like it was done in the early history of calculus)
The only catch about SIA is that it requires you to use intuitionistic logic rather than classical logic in your mathematical arguments (which I admit is a barrier, but it also buys you some things). And what it offers is much simpler proofs that support intuitive reasoning.
There is also this book, "A Primer of Infinitesimal Analysis" [0], which develops a big chunk of calculus and classical mechanics using only infinitesimals, and is fully rigorous.
I wonder if it's working with floating point numbers that made me less uncomfortable when first discovering infinitesimals. The idea that something just falls out of our current representable scope under certain operations seemed fine to me. I've always had a soft spot for infinitesimals and a slight dislike for epsilon-delta limits.
I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.)
[0] https://en.wikipedia.org/wiki/The_Analyst