[Enginerrrd]

>One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals.

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.

[tgflynn]

So what practical applications do these functions have whose derivatives can't be approximated numerically ?

[eximius]

Probably lots in physics, aerospace, electrical engineering.

And, again, coming up with derived theorems that are useful.