It is often useful to describe the behavior of a
function as either the
argument or the function value gets "infinitely large" in some sense. For example, consider the function defined by
The
graph of this function has a horizontal
asymptote at Geometrically, when moving increasingly farther to the right along the -axis, the value of approaches0. This limiting behavior is similar to the
limit of a function in which the
real number approaches except that there is no real number to which approaches.
By adjoining the elements and to it enables a formulation of a "limit at infinity", with
topological properties similar to those for
To make things completely formal, the
Cauchy sequences definition of allows defining as the set of all
sequences of
rational numbers such that every is associated with a corresponding for which for all The definition of can be constructed similarly.
Measure and integration
In
measure theory, it is often useful to allow sets that have infinite
measure and integrals whose value may be infinite.
Such measures arise naturally out of calculus. For example, in assigning a measure to that agrees with the usual length of
intervals, this measure must be larger than any finite real number. Also, when considering
improper integrals, such as
the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as
The extended real number system , defined as or , can be turned into a
totally ordered set by defining for all With this
order topology, has the desirable property of
compactness: Every
subset of has a
supremum and an
infimum[3] (the infimum of the
empty set is , and its supremum is ). Moreover, with this
topology, is
homeomorphic to the
unit interval Thus the topology is
metrizable, corresponding (for a given homeomorphism) to the ordinary
metric on this interval. There is no metric, however, that is an extension of the ordinary metric on
In this topology, a set is a
neighborhood of if and only if it contains a set for some real number The notion of the neighborhood of can be defined similarly. Using this characterization of extended-real neighborhoods,
limits with tending to or , and limits "equal" to and , reduce to the general topological definition of limits—instead of having a special definition in the real number system.
Arithmetic operations
The arithmetic operations of can be partially extended to as follows:[2]
When dealing with both positive and negative extended real numbers, the expression is usually left undefined, because, although it is true that for every real nonzero sequence that
converges to the
reciprocal sequence is eventually contained in every neighborhood of it is not true that the sequence must itself converge to either or Said another way, if a
continuous function achieves a zero at a certain value then it need not be the case that tends to either or in the limit as tends to This is the case for the limits of the
identity function when tends to and of (for the latter function, neither nor is a limit of even if only positive values of are considered).
However, in contexts where only non-negative values are considered, it is often convenient to define For example, when working with
power series, the
radius of convergence of a power series with
coefficients is often defined as the reciprocal of the
limit-supremum of the sequence . Thus, if one allows to take the value then one can use this formula regardless of whether the limit-supremum is or not.
Algebraic properties
With these definitions, is not even a
semigroup, let alone a
group, a
ring or a
field as in the case of However, it has several convenient properties:
and are either equal or both undefined.
and are either equal or both undefined.
and are either equal or both undefined.
and are either equal or both undefined
and are equal if both are defined.
If and if both and are defined, then
If and and if both and are defined, then
In general, all laws of arithmetic are valid in —as long as all occurring expressions are defined.
Miscellaneous
Several functions can be
continuouslyextended to by taking limits. For instance, one may define the extremal points of the following functions as:
Some
singularities may additionally be removed. For example, the function can be continuously extended to (under some definitions of continuity), by setting the value to for and for and On the other hand, the function cannot be continuously extended, because the function approaches as approaches from below, and as approaches from above, i.e., the function not converging to the same value as its independent variable approaching to the same domain element from both the positive and negative value sides.
A similar but different real-line system, the
projectively extended real line, does not distinguish between and (i.e. infinity is unsigned).[5] As a result, a function may have limit on the projectively extended real line, while in the extended real number system only the
absolute value of the function has a limit, e.g. in the case of the function at On the other hand, on the projectively extended real line, and correspond to only a limit from the right and one from the left, respectively, with the full limit only existing when the two are equal. Thus, the functions and cannot be made continuous at on the projectively extended real line.
^Some authors use Affinely extended real number system and Affinely extended real number line, although the extended real numbers do not form an
affine line.
^Read as "positive infinity" and "negative infinity" respectively.