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Calculus: a limitless perspective

Michael P. Lamoureux, Matt Yedlin

TL;DR

The paper tackles the difficulty students face with limits by proposing a limit-free foundation for calculus based on controlled approximations and error functions. It defines continuity and differentiability through first-order approximations, formalizes an algebra of error functions, and builds calculus rules (sum, product, chain, inverse, L'Hôpital, Taylor) via error-based reasoning. It then demonstrates geometry-guided constructions of the trigonometric, hyperbolic, and exponential functions, deriving their derivatives directly from area-based definitions, and shows how Taylor and FTC arise in this framework. The approach promises a more intuitive yet rigorous introduction to calculus, grounded in geometry and linear algebra, with potential to ease cognitive load while preserving mathematical power and coherence.

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'Hôpital's Rule, Taylor polynomials, and the Fundamental Theorem of Calculus, all via approximation.

Calculus: a limitless perspective

TL;DR

The paper tackles the difficulty students face with limits by proposing a limit-free foundation for calculus based on controlled approximations and error functions. It defines continuity and differentiability through first-order approximations, formalizes an algebra of error functions, and builds calculus rules (sum, product, chain, inverse, L'Hôpital, Taylor) via error-based reasoning. It then demonstrates geometry-guided constructions of the trigonometric, hyperbolic, and exponential functions, deriving their derivatives directly from area-based definitions, and shows how Taylor and FTC arise in this framework. The approach promises a more intuitive yet rigorous introduction to calculus, grounded in geometry and linear algebra, with potential to ease cognitive load while preserving mathematical power and coherence.

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'Hôpital's Rule, Taylor polynomials, and the Fundamental Theorem of Calculus, all via approximation.
Paper Structure (39 sections, 126 equations, 18 figures)

This paper contains 39 sections, 126 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Introducing approximations
  3. The algebra of error functions
  4. Definition of continuity
  5. Combining continuous functions
  6. Definition of derivative
  7. The derivative is unique
  8. Derivative of a monomial
  9. Derivative rules
  10. Chain rule, visually
  11. Rational powers of x
  12. Demonstration: Calculus from geometry
  13. The trigonometric functions
  14. Defining cosine and sine for small arguments
  15. Extending to large arguments
  16. ...and 24 more sections

Figures (18)

  • Figure 1: Bounding box on an error function, demonstrating Property 3.
  • Figure 2: A sequence of bounding boxes, demonstrating Property 3.
  • Figure 3: A funnel to visualize property 3 of the error function.
  • Figure 4: Composing two functions and the chain rule.
  • Figure 5: Three curves $x^2 + y^2 = 1$, $x^2-y^2 = 1$, $xy=1$, and associated regions defining trigonometric, hyperbolic, and exponential functions.
  • ...and 13 more figures