# Limit

## Applications

The limit concept is essential to understanding the real number system and its distinguishing characteristics. In one sense real numbers can be defined as the numbers that are the limits of convergent sequences of rational numbers. One application of the concept of limits is on the derivative. The derivative is a rate of flow or change, and can be computed based on some limits concepts. Limits are also key to calculating intergrals (expressions of areas). The integral calculates the entire area of a region by summing up an infinite number of small pieces of it. Limits are also part of the iterative process. An iteration is repeatedly performing a routine, using the output of one step as the input of the next step. Each output is an iterate. Some successful iterates can get as close as desired to a theoretically exact value.

## Resources

### Books

Abbot, P., and M.E. Wardle. Teach Yourself Calculus. Lincolnwood: NTC Publishing, 1992.

Allen, G.D., C. Chui, and B. Perry. Elements of Calculus. 2nd ed. Pacific Grove: Brooks/Cole Publishing Co., 1989.

Gowar, Norman. An Invitation to Mathematics. New York: Oxford University Press, 1979.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Silverman, Richard A. Essential Calculus With Applications. New York: Dover, 1989.

### Periodicals

McLaughlin, William I. "Resolving Zeno's Paradoxes." Scientific American 271 (1994): 84-89.

## KEY TERMS

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Converge

—To converge is to approach a limit that has a finite value.

Interval

—An interval is a subset of the real numbers corresponding to a line segment of finite length, and including all the real numbers between its end points. An interval is closed if the endpoints are included and open if they are not.

Real Number

—The set of numbers containing the integers and all the decimals including both the repeating and nonrepeating decimals.

Sequence

—A sequence is a series of terms, in which each successive term is related to the one before it by a fixed formula.