IMO, the calculus is taught incorrectly. It should start with functions and completely avoid sequences initially. Once you understand how calculus exploits continuity (and sometimes smoothness), it becomes almost intuitive. That's also how it was historically developed, until Weierstrass invented his monster function and forced a bit more rigor.

But instead calculus is taught from fundamentals, building up from sequences. And a lot of complexity and hate comes from all those "technical" theorems that you need to make that jump from sequences to functions. E.g. things like "you can pick a converging subsequence from any bounded sequence".

Interesting.

In Maths classes, we started with functions. Functions as list of pairs, functions defined by algebraic expressions, functions plotted on graph papers and after that limits. Sequences were peripherally treated, just so that limits made sense.

Simultaneously, in Physics classes we were being taught using infinitesimals, with the a call back that "you will see this done more formally in your maths classes, but for intuition, infinitesimals will do for now".