Calculus: a limitless perspective

Abstract

We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with a linear function. The errors in approximation are defined as a class of functions with certain properties; rules for combining error functions lead to all the familiar results in differential calculus. We believe that this approach is more natural for students while still giving a rigourous foundation to differential calculus. We demonstrate its utility by deriving the basic differential rules for trigonometric, hyperbolic and exponential functions, as well as L'H\^opital's Rule, Taylor polynomials, and the Fundamental Theorem of Calculus, all via approximation.