Figure 2.1: Isaac Newton (1642-1727). This statue of Isaac Newton in Cambridge University was described in The Prelude, a poem by William Wordsworth (1770–1850):“Newton with his prism and silent face, The marble index of a mind for ever Voyaging through strange seas of Thought, alone.”

Calculus is usually divided into two branches, differential and integral, partly for historical reasons. The subject grew out of efforts in the seventeenth century to solve two important problems: finding instantaneous rates of change, e.g. velocity (differential calculus), and computing areas under curves (integral calculus). Isaac Newton and Gottfried Wilhelm Leibniz are generally considered to be the co-founders of Calculus.

This chapter introduces the limit concept and sets the stage for our understanding of instantaneous velocity and our study of the derivative in Chapter 3. The first section, intended as motivation, discusses how limits arise in the study of rates of change and tangent lines.