I actually love the intuition built with geometric arguments using infinitesimals.
You need something like smooth infinitesimal analysis [0] to make this rigorous, but it's much better than anything involving limits.
You need something like smooth infinitesimal analysis [0] to make this rigorous, but it's much better than anything involving 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.)
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.
[0] https://www.cambridge.org/br/universitypress/subjects/mathem...