Or you can hand wave a bit and trust intuition. Just like the titans who invented it all did!
The obsession with rigor that later developed -- while necessary -- is really an "advanced topic" that shouldn't displace learning the intuition and big picture concepts. I think math up through high school should concentrate on the latter, while still being honest about the hand-waving when it happens.
I broadly agree. But, the big risk here is that it's really easy for an adventurous student to stretch that handwaving beyond where it's actually valid. You at least have to warn them that the "intuitions" you give them are not general methods, just explanations for why the algorithms you teach them do something worthwhile (and for the ones inclined to explore, give them some fun edge cases to think about).
You can do it with synthetic differential geometry, but that introduces some fiddliness in the underlying logic in order to cope with the fact that eps^2 really "equals" zero for small enough eps, and yet eps is not equal to zero.
calculus works... because it was almost designed for Mechanics. If the machine it's getting input, you have output. When it finished getting input, all the output you get yields some value, yes, but limits are best understood not for the result, but for the process (what the functions do).
You are not sending 0 coins to a machine, do you? You sent X to 0 coins to a machine. The machine will work from 2 to 0, but 0 itself is not included because is not a part of a changing process, it's the end.
Limits are for ranges of quantities over something.
