Limits and Derivatives

Abstract

Over a period of 350 years or more, calculus has evolved conceptually and in notation. Up until recently, calculus was described using infinitesimals, which are numbers so small, they can be ignored in certain products. This led to arguments about “ratios of infinitesimally small quantities” and “ratios of evanescent quantities”. Eventually, it was the French mathematician Augustin-Louis Cauchy (1789–1857), and the German mathematician Karl Weierstrass (1815–1897), who showed how limits can replace infinitesimals. However, in recent years, infinitesimals have bounced back onto the scene in the field of “non-standard analysis”, pioneered by the German mathematician Abraham Robinson (1918–1974). Robinson showed how infinitesimal and infinite quantities can be incorporated into mathematics using simple arithmetic rules:

$$\begin{aligned} \text {infinitesimal} \times \text {bounded} &= \text {infinitesimal} \\ \text {infinitesimal}\times \text {infinitesimal}&=\text {infinitesimal} \end{aligned}$$

where a bounded number could be a real or integer quantity. So, even though limits have been adopted by modern mathematicians to describe calculus, there is still room for believing in infinitesimal quantities.