[enriquto]

You can use nilpotents with classical logic and the excluded middle. This is called dual numbers and it's already a good model for "calculus without limits". They are like complex numbers, but instead of x^2=-1 you set x^2=0.

However, if you want to get really serious about that, you'll need that zero plus an infinitessimal be equal to zero. This is impossible in classical logic due to the excluded middle (which forces each number to be either equal to zero or non-zero).

[Qem]

Can you recommend a introductory calculus book that builds it up from dual numbers?

[enriquto]

The Silvanus P. Thompson book suggested by the sibling comment is lovely and very clear.

For a more algebraic treatment, and its important applications to automatic differentiation, I'd suggest starting with the relevant wikipedia articles:

https://en.wikipedia.org/wiki/Dual_number

https://en.wikipedia.org/wiki/Automatic_differentiation

[zozbot234]

You could try Calculus made easy by S. P. Thompson.