Chapter 2: Limits and Derivatives

Students learn about the formal definition of a limit and the distinction between finite limits and infinite limits, as well as limits at infinity. The chapter emphasizes the importance of limits in defining continuity, showing how removable, jump, and infinite discontinuities arise in different types of functions. Special limit laws are introduced, including sum, difference, product, quotient, and power rules, alongside strategies for evaluating limits algebraically. Indeterminate forms such as 0/0 and ∞/∞ are explained, preparing readers for L’Hôpital’s Rule in later chapters. The section on tangent lines and velocities bridges limits with real-world applications, illustrating how the slope of a tangent line represents an instantaneous rate of change. Using the idea of a secant line approaching a tangent line, the chapter introduces the derivative informally as the limit of the average rate of change. Students also explore the connection between limits and motion, modeling position, velocity, and acceleration in the context of calculus. By the end of the chapter, learners understand how limits form the mathematical foundation for derivatives and integrals, enabling precise analysis of change in both physical and abstract systems.