Calculus Workbook For Dummies with Online Practice
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When you work with limit and continuity problems in calculus, there are a couple of formal definitions you need to know about. So, before you take on the following practice problems, you should first re-familiarize yourself with these definitions.

Here is the formal, three-part definition of a limit:

For a function f (x) and a real number a,

The limit of a function when x reaches a.

exists if and only if

The three-part definition of a limit

(Note that this definition does not apply to limits as x approaches infinity or negative infinity.)

Now, here's the definition of continuity:

A function f (x) is continuous at a point a if three conditions are satisfied:

The three conditions that make a function f (x) continuous

Now it's time for some practice problems.

Practice questions

Using the definitions and this figure, answer the following questions.

Graph of a function with many discontinuities.
A function with many discontinuities.
  1. At which of the following x values are all three requirements for the existence of a limit satisfied, and what is the limit at those x values?

    x = –2, 0, 2, 4, 5, 6, 8, 10, and 11.

  2. At which of the x values are all three requirements for continuity satisfied?

Answers and explanations

  1. All three requirements for the existence of a limit are satisfied at the x values 0, 4, 8, and 10:

    At 0, the limit is 2.

    At 4, the limit is 5.

    At 8, the limit is 3.

    At 10, the limit is 5.

    To make a long story short, a limit exists at a particular x value of a curve when the curve is heading toward some particular y value and keeps heading toward that y value as you continue to zoom in on the curve at the x value. The curve must head toward that y value (that height) as you move along the curve both from the right and from the left (unless the limit is one where x approaches infinity).

    The phrase heading toward is emphasized here because what happens precisely at the given x value isn't relevant to this limit inquiry. That's why there is a limit at a hole like the ones at x = 8 and x = 10.

  2. The function in the figure is continuous at 0 and 4.

    The common-sense way of thinking about continuity is that a curve is continuous wherever you can draw the curve without taking your pen off the paper. It should be obvious that that's true at 0 and 4, but not at any of the other listed x values.

About This Article

This article is from the book:

About the book author:

Mark Ryan has taught pre-algebra through calculus for more than 25 years. In 1997, he founded The Math Center in Winnetka, Illinois, where he teaches junior high and high school mathematics courses and standardized test prep classes. He also does extensive one-on-one tutoring. He is a member of the Authors Guild and the National Council of Teachers of Mathematics. Ryan is the author of Calculus For Dummies, Calculus Workbook For Dummies, and Geometry Workbook For Dummies.

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