The concept of a limit is central to calculus, and also plays a significant role in other branches of mathematics. Calculus has two major subfields: differential calculus and integral calculus. Differential calculus is concerned with the rate of change and finding tangent lines to curves. Integral calculus is about computing the total area under a curve and measuring the total effect of a process of continuous change. These concepts are defined in terms of limits. In fact, every single notion of calculus is a limit in one way or another.
The concept of a limit was certainly not the basis of differential and integral calculus when they were first developed. This lack of rigor led to criticisms of calculus at the early stages. The implicit usage of limits can be traced back to the works of the ancient Greek scholars. Isaac Newton, in his renowned Mathematical Principles of Natural Philosophy (1686–1687), introduced his method of first and last ratios, which foreshadowed the development of the theory of limits. However, none of the prominent mathematicians of the 18th century attempted to establish calculus on the concept of limit, thus addressing the valid criticisms that the calculus faced at that time. Euler's perspective on this issue is noteworthy: in the introduction to his treatise on Differential Calculus (1755), he mentions the concept of limit, but does not use it throughout the book. A significant change in this perspective came with Augustin-Louis Cauchy's Algebraic Analysis (1821) and subsequent publications, in which the theory of limits was first developed. Cauchy used this concept as a tool to establish a rigorous foundation for mathematical analysis. His approach, which demystified the foundations of analysis and calculus, was widely accepted.
In the 1960s an alternative formulation of calculus, called non-standard analysis, was introduced. Non-standard analysis legitimizes the concept of infinitesimals, which was vaguely used by Gottfried Wilhelm Leibniz (1646–1716), Leonhard Euler (1707–1783), and many others in the beginnings of calculus.
In this chapter, we introduce the concepts of limit and continuity first intuitively and simply, and then more rigorously. We will learn how to evaluate many limits, but one of the major techniques for finding limits will be discussed when we talk about applications of differentiation.
