Extended real number line

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In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (read as "positive infinity" and "negative infinity," respectively). These new elements are not real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affinely extended real number system is denoted R or [−∞, +∞]. The affinely extended real number system should be distinguished from the projectively extended real numbers which has only one infinity, rather than two.

When the meaning is clear from context, the symbol +∞ is often written simply as ∞.

Contents

Motivation

Limits

We often wish to describe the behavior of a function f(x), as either the argument x or the function value f(x) gets "very big" in some sense. For example, consider the function

The graph of this function has a horizontal asymptote at f(x) = 0. Geometrically, as we move farther and farther to the right along the x-axis, the value of 1/x2 approaches 0. This limiting behavior is similar to the limit of a function at a real number, except that there is no real number to which x approaches.

By adjoining the elements +∞ and −∞ to R, we allow a formulation of a "limit at infinity" with topological properties similar to those for R.

To make things completely formal, the Cauchy sequences definition of R allows us to define +∞ as the set of all sequences of rationals which, for any K>0, from some point on exceed K. We can define −∞ similarly.

Measure and integration

In measure theory, it is often useful to allow sets which have infinite measure and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure to R that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering infinite integrals, such as

the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as

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