Page 47 - Calculus Workbook For Dummies

Chapter 3

A Graph Is Worth a Thousand Words:

Limits and Continuity

In This Chapter

The mathematical mumbo jumbo of limits and continuity
When limits exist and don’t exist
Discontinuity. . . or graphus interruptus

Y ou can use ordinary algebra and geometry when the things in a math problem aren’t

changing (sort of) and when lines are straight. But you need calculus when things are
changing (these changing things are often represented as curves). For example, you need
calculus to analyze something like the motion of the space shuttle during the beginning of
its flight because its acceleration is changing every split second.

Ordinary algebra and geometry fall short for such things because the algebra or geometry
formula that works one moment no longer works a millionth of a second later. Calculus, on
the other hand, chops up these constantly changing things — like the motion of the space
shuttle — into such tiny bits (actually infinitely small bits) that within each bit, things don’t
change. Then you can use ordinary algebra and geometry.

Limits are the “magical” trick or tool that does this chopping up of something into infinitely
small bits. It’s the mathematics of limits that makes calculus work. Limits are so essential
that the formal definitions of the derivative and the definite integral both involve limits.
If — when your parents or other adults asked you, “What do you want to be when you grow
up?” — you responded, “Why, a mathematician, of course,” then you may ultimately spend a
great deal of time thoroughly studying the deep and rich subtleties of continuity. For the rest
of you, the concept of continuity is a total no-brainer. If you can draw a graph without lifting
your pen or pencil from the page, the graph is continuous. If you can’t — because there’s a
break in the graph — then the graph is not continuous. That’s all there is to it. By the way,
there are some subtle and technical connections between limits and continuity (which I
don’t want to get into), and that’s why they’re in the same chapter. But, be honest now,
did you buy this book because you were dying to learn about mathematical subtleties and
technicalities?

Digesting the Definitions: Limit and Continuity

This short section covers a couple formal definitions and a couple other things you need to
know about limits and continuity. Here’s the formal, three-part definition of a limit: