> And you can't easily only partially include infinity.
Sure you can. You just need to use a dx that's small enough for the particular functions you're working with and the degree of precision you need.
One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals.
The later, at least, has been my experience. I think I have a fairly decent practical intuition for calculus based on imagining dx becoming smaller and smaller until it's small enough, but I don't think my brain has any actual representation of "true infinitesimals" and my intuition breaks down completely if I try to imagine things like the relationship between the rational and irrational numbers. Maybe that's due to my intellectual limitations, but I wonder if it isn't because these concepts might be over-elaborate abstractions that don't really exist in our world.
Don't derivatives count? That's a pretty important and trivial calculation. Sure, you can approximate it when it's nicely behaved, but they aren't always. There's also lots of verrrrrrry slowly converging series that can't be easily computed numerically.
And, again, coming up with derived theorems that are useful.
This is like Stacey King saying "I'll always remember this as the night that Michael Jordan and I combined for 70 points."
Pretty much the entire field of Analysis (of which calculus is a part) relies on 'infinities' of some kind. Even if you try to restrict to the rationals, you're still typically working with infinite series of them.
You can do some analysis over the rationals, but often this takes the form of Cauchy sequences of rationals which might be cheating.
[0] https://math.stackexchange.com/questions/387234/how-far-can-...
And you can't easily only partially include infinity.