What is Calculus?: Limits

This is the second in a series of short articles exploring calculus. The first article explored the origins of calculus. The next few articles explore the mathematics of calculus. This article focuses on the foundation of calculus: limits.

The first article in this series asked the question: What is calculus? I promised then that the second article in the series would explore the substance of calculus, the mathematics of calculus. So let’s dive right in.

Here’s my two-part answer to the “what is calculus?” question:

Calculus is a mindset—a dynamics mindset.
Contentwise, calculus is the
mathematics of infinitesimal change.

The second sentence describes the mathematics of calculus. But I don’t expect you to understand that sentence just yet. That’s where the first sentence comes in. If you ask me, all of calculus flows from a more fundamental—and intuitive—principle, articulated in the first sentence: the notion that calculus is a dynamics mindset. Let me explain.

Calculus: A New Way of Thinking

The mathematics that precedes calculus—often called “pre-calculus,” which includes algebra and geometry—largely focuses on static problems: problems devoid of change. By contrast, change is central to calculus. Calculus is all about dynamics. Example:

  • What’s the perimeter of a square of side length 2 feet? ← Pre-calculus problem.
  • How fast is the square’s perimeter changing if its side length is increasing at the constant rate of 2 feet per second? ← Calculus problem.

Now that I’ve sensitized you to thinking “calculus!” whenever you read about or infer the presence of change, take a quick look at the second sentence in my two-part answer above. What’s the last word? Change. But it’s a new type of change—infinitesimal change—and this requires some explaining. That’s our next stop.

A Philosopher Walks Into a Starbucks

First, a rough definition of “infinitesimal change”:

“Infinitesimal change” means: as close to zero change as you can imagine, but not zero change.

“What?!” I hear you saying. So let me illustrate this definition via my friend, Zeno of Elea (c. 490-430 BC). This ancient Greek philosopher thought up a set of paradoxes arguing that motion is not possible. One such paradox—the Dichotomy Paradox—can be stated as follows:

To travel a certain distance you must first traverse half of it.

Makes perfect sense. Two is one plus one. And one is one-half plus one-half. But don’t be fooled by this seemingly innocent reasoning; it’s a trap! (Admiral Ackbar!) 

To appreciate what’s going on—and connect Zeno’s paradox back to calculus—let’s pretend Zeno is in line at Starbucks, two feet away from the cash register. He’s almost done scanning the menu when the barista yells out “next!” And that’s when poor Zeno panics. He must now walk two feet, but because of his mindset, he walks only half that distance with his first step. He then walks half of the remaining distance with his second step. (Can you imagine how annoying those in line behind him are getting?) The figure below keeps track of the total distance d Zeno has walked and the change in distance Δd after each of his steps.

Fig. 1.2: Zeno trying to walk a distance of 2 feet by traversing half the remaining distance with each step. (Reprinted, with permission, from Calculus Simplified.)

Here’s a tabular representation of the action:

Table 1.1: The distance d and change in distance Δd after each of Zeno’s steps. (Reprinted, with permission, from Calculus Simplified.)

Each change d in Zeno’s distance is half the previous one. So as Zeno continues his walk, d gets closer to zero but never becomes zero.[1] If we checked back in with Zeno after he’s taken an infinite

amount of steps—what a patient barista!—the change d resulting from his next step would be . . . drum roll please . . . an infinitesimal change—as close to zero as you can imagine but not equal to zero.

This example, in addition to illustrating what an infinitesimal change is, also does two more things. First, it illustrates the dynamics mindset of calculus. We discussed Zeno walking; we thought about the change in the distance he traveled; we visualized the situation with a figure and a table that each conveyed movement. (Calculus is full of action verbs!) Second, the example challenges us. Clearly, one can walk 2 feet. But as Table 1.1 suggests, that doesn’t happen during Zeno’s walk—he approaches the 2-foot mark with each step yet never arrives. How do we describe this fact with an equation? (That’s the challenge.) No pre-calculus equation will do. We need a new concept that quantifies our very dynamic conclusion. That new concept is the mathematical foundation of calculus: limits.

Limits: The Foundation of Calculus

In modern calculus speak we paraphrase the main takeaway of Table 1.1 this way: the distance d traveled by Zeno approaches 2 as Δd approaches zero. It’s important to note that d never equals 2 and Δd never equals 0. Today we express these conclusions more compactly by writing

read “the limit of d as Δd approaches zero (but is never equal to zero) is 2.” This new equation—and what we take it to mean—remind us that d is always approaching 2 yet never arrives at 2. (Oh, the dynamics!) The same idea holds for Δd: it is always approaching 0 yet never arrives at 0. Said more succinctly:

Limits approach indefinitely (and thus never arrive).

You’ve now met the foundational concept of calculus—limit. You’ve also gotten a glimpse of what infinitesimal change means and how a limit encodes that notion. Finally, you’ve seen many times how a dynamics mindset is at the core of calculus’ new way of thinking about mathematics. In the next article in this series we’ll employ a dynamics mindset and limits to solve the three Big Problems that drove the development of calculus—instantaneous speed, the tangent line problem, and the area problem (discussed in the first post in this series). See you then!

Footnote: [1] Because each d is always half of a positive number.

 

Calculus Simplified
By Oscar E. Fernandez

Calculus is a beautiful subject that most of us learn from professors, textbooks, or supplementary texts. Each of these resources has strengths but also weaknesses. In Calculus Simplified, Oscar Fernandez combines the strengths and omits the weaknesses, resulting in a “Goldilocks approach” to learning calculus: just the right level of detail, the right depth of insights, and the flexibility to customize your calculus adventure.

Fernandez begins by offering an intuitive introduction to the three key ideas in calculus—limits, derivatives, and integrals. The mathematical details of each of these pillars of calculus are then covered in subsequent chapters, which are organized into mini-lessons on topics found in a college-level calculus course. Each mini-lesson focuses first on developing the intuition behind calculus and then on conceptual and computational mastery. Nearly 200 solved examples and more than 300 exercises allow for ample opportunities to practice calculus. And additional resources—including video tutorials and interactive graphs—are available on the book’s website.

Calculus Simplified also gives you the option of personalizing your calculus journey. For example, you can learn all of calculus with zero knowledge of exponential, logarithmic, and trigonometric functions—these are discussed at the end of each mini-lesson. You can also opt for a more in-depth understanding of topics—chapter appendices provide additional insights and detail. Finally, an additional appendix explores more in-depth real-world applications of calculus.

Learning calculus should be an exciting voyage, not a daunting task. Calculus Simplified gives you the freedom to choose your calculus experience, and the right support to help you conquer the subject with confidence.

  • An accessible, intuitive introduction to first-semester calculus
  • Nearly 200 solved problems and more than 300 exercises (all with answers)
  • No prior knowledge of exponential, logarithmic, or trigonometric functions required
  • Additional online resources—video tutorials and supplementary exercises—provided
This post is part of a series, explore additional posts here<< What is Calculus?What is Calculus?: The Two Pillars >>