Can we operate on the real numbers in calculus?

Question

For a set theory class, I was reading into the definition and properties of real numbers. Real numbers are Archimedean. That means there are no infinitely large real numbers or infinitesimally small real numbers. However, the concept of integrals and derivatives seem fundamentally tied to the concept of infinitesimal values. How can we preform calculus on the reals, if there are no infinitesimally small real numbers?

Answer 1

That is the basic question of calculus. If you've ever learned calculus, then you've seen the answer: limits work in place of infinitesimals. People used infinitesimals instead for centuries, but later people were dissatisfied with the rigor of this-but it's all the same, really. They're neither necessary nor contradictory.

Answer 2

One major misconception about the calculus is that there are two approaches: limits and infinitesimals. The truth is that there are two approaches: Leibniz's approach with infinitesimals, and Weierstrass's paraphrase thereof using epsilon-delta using an Archimedean continuum. The concept of limit is present in both approaches. Thus, the limit of a sequence $(u_n)$ will be the value of $u_n$ for infinite $n$, rounded off to the nearest real (i.e., take the standard part).