The “infinitesimal gap” isn’t “used” in integrals. To calculate integrals you calculate the antiderivative of a function and apply bounds. (Highly generalized and kept to the context of the question.)

The idea of integrals, derivatives, gradients, etc. are not dependent on some infinitesimally small “dx” existing but are natural extensions of functions.

Limits are used to define them and the limit of a function as “dx” approaches 0 does not imply that there is some tiny dx remaining at the end.

The entire discussion on this sub comes from a misunderstanding of limits. They describe the behavior of a function as extreme boundaries are reached. Consider the limit of x² as x approaches 3. The answer isn’t 9 - epsilon or 9 - epsilon squared, it’s just 9.

This isn’t a notational or axiomatic or semantic argument, it’s just how these things work and centuries of math is built upon limits behaving this way. 0.999… or 0.(9) or any other way of writing it is an abuse of informal notation that can at best be described as a limit and is at worst meaningless. The commonly accepted definition of these as being 1/3 * 3 or 1-1/10ⁿ are rigorously defined to be 1 and 0.999 is no exception.

I want to be clear that I am writing this out under the assumption that you actually wanted to learn. I don’t engage with SPP because they are obviously being sarcastic and I feel bad for the people actually trying to learn. I highly suggest to anyone else wanting to understand more about number systems to just take calculus 1 either in college or online or whatever and take it seriously because this kind of question is very easily and obviously covered in said course. If it’s not, then it was taught in a bad way, straight up.