Historically, a full understanding of just what a limit was didn't occur until some 200 years after Newton and Leibniz used a naive notion of the limit to develop the calculus of differentiation and integration. Modern calculus texts believe in the logical progression that starts from a complete understanding of the limit so that the concepts of the derivative and the integral are well founded. But all of the applications of differentiation and integration that were developed by the giants of 17th through the 19th centuries were obtained while the struggle to understand the limit was ongoing. The missing ingredient in this struggle ultimately turned out to be a complete understanding of the real number system.

It would be entirely possible to learn about the derivative and its applications with a naive understanding of a limit, and the manipulative skills needed to apply the formal rules for calculating limits. This approach can be taken with this Study Guide because it is supported by Maple, which can provide all the limits needed for developing and exploring the formalism of the calculus. On the other hand, logical purists are welcomed to begin with a rigorous understanding of the limit, as per these development in Chapter 1.