That is fine for you, not everyone needs to know about Wiener path integral formulation of quantum mechanics ... which is based on functions that are almost continuous everywhere but nowhere differentiable (the set of functions that are differentiable and with which you are familiar is a set of measure zero i.e. it is comparable to using only integers instead of real numbers). The limit process, which you discard as not meaning anything, is essential to understand the Wiener path integral formalism. Fortunately, mathematicians and physicists did not "lose sight of that" ... and are able to go beyond what your limited understanding would limit you to. I don't mean this in a deragotary way - you obviously don't need to understand at a deeper level than you do; few people do ... but to those that need this deeper level, the mathematical formalism that you dismiss is, in fact, essential.
Edit: my apology for making an unwarranted assumption about what your level of understanding is.
I don't mean this in a deragotary way - you obviously don't need to understand at a deeper level than you do; few people do ...
Do you like to assume much?
I have a masters in mathematics. I likely understand a lot more than you give me credit for.
I do not "discard" the limit process. I'm saying that the d/dx notation, which we get from the old infinitesmal approach, has insights buried in it that most students miss. I certainly missed it for a long time.
Limits won for good reason - they were the first consistent approach to defining the derivative that mathematicians understood, and were a key step on the way to resolving the inconsistencies in previous definitions that Fourier series had revealed.
But limits are not actually "essential". I am personally aware of multiple completely rigorous formalizations of Calculus. The second most famous of which is Robinson's non-standard analysis, which has absolutely nothing looking like a limit in sight.
Edit: my apology for making an unwarranted assumption about what your level of understanding is.