# Limit - History, Limit Of A Sequence, Limit Of A Function, Applications

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limits real based nearness

In **mathematics** the concept of limit formally expresses the notion of arbitrary closeness. That is, a limit is a value that a **variable** quantity approaches as closely as one desires. The operations of differentiation and integration from **calculus** are both based on the theory of limits. The theory of limits is based on a particular property of the **real numbers**; namely that between any two real numbers, no matter how close together they are, there is always another one. Between any two real numbers there are always infinitely many more.

Nearness is key to understanding limits: only after nearness is defined does a limit acquire an exact meaning. Relevantly, a neighborhood of points near any given point comprise a neighborhood. Neighborhoods are definitive components of infinite limits of a sequence.

## Additional Topics

The ancient Greek philosopher Zeno may have been one of the first mathematicians to ponder the limit of a sequence and wonder how it related to the world around him. Zeno argued that all motion was impossible because in order to move a distance l it is first necessary to travel half the distance, then half the remaining distance, then half of that remaining distance and so on. Thus, he argued, the…

Consider an arbitrary function, y = f(x). (A function is a set of ordered pairs for which the first and second elements of each pair are related to one another in a fixed way. When the elements of the ordered pairs are real numbers, the relationship is usually expressed in the form of an equation.) Suppose that successive values of x are chosen to match those of a converging sequence such as the s…

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…

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## User Comments

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