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Chapter 1 Limits

The foundation of “the calculus” is the limit. It is a tool to describe a particular behavior of a function. This chapter begins our study of the limit by approximating its value graphically and numerically. After a formal definition of the limit, properties are established that make “finding limits” tractable. Once the limit is understood, then the problems of area and rates of change can be approached.

Figure 1.0.1. Overview of Calculus

Chapter Summary.

In this chapter we:

  • defined the limit,

  • found accessible ways to approximate their values numerically and graphically,

  • developed a not-so-easy method of proving the value of a limit (\(\varepsilon\)-\(\delta\) proofs),

  • explored when limits do not exist,

  • defined continuity and explored properties of continuous functions, and

  • considered limits that involved infinity.

Why? Mathematics is famous for building on itself and calculus proves to be no exception. In the next chapter we will be interested in “dividing by \(0\text{.}\)” That is, we will want to divide a quantity by a smaller and smaller number and see what value the quotient approaches. In other words, we will want to find a limit. These limits will enable us to, among other things, determine exactly how fast something is moving when we are only given position information.

Later, we will want to add up an infinite list of numbers. We will do so by first adding up a finite list of numbers, then take a limit as the number of things we are adding approaches infinity. Surprisingly, this sum often is finite; that is, we can add up an infinite list of numbers and get, for instance, \(42\text{.}\)

These are just two quick examples of why we are interested in limits. Many students dislike this topic when they are first introduced to it, but over time an appreciation is often formed based on the scope of its applicability.