We often wish to describe the behavior of a function , as either the argument or the function value gets "very big" in some sense. For example, consider the function
The graph of this function has a horizontal asymptote at y = 0. Geometrically, as we move farther and farther to the right along the -axis, the value of 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 approaches.
By adjoining the elements and to , we allow 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 us to define as the set of all sequences of rationals which, for any , from some point on exceed . We can define similarly.
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
This induces the order topology on . In this topology, a set is a neighborhood of if and only if it contains a set for some real number , and analogously for the neighborhoods of . is a compactHausdorff spacehomeomorphic 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 that is an extension of the ordinary metric on .
With this topology the specially defined limits for tending to and , and the specially defined concepts of limits equal to and , reduce to the general topological definitions of limits.
The expression is not defined either as or , because although it is true that whenever for a continuous function it must be the case that is eventually contained in every neighborhood of the set , it is not true that must tend to one of these points. An example is which is of the form but does not tend to either or when . For instance, but does not exist because but . (The modulus, nevertheless, does approach .)
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 . The function can not be continuously extended because the function approaches as approaches 0 from below, and as approaches from above.
Compare the projectively extended real line, which does not distinguish between and . As a result, on one hand a function may have limit on the projectively extended real line, while in the affinely 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
correspond on the projectively extended real line 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 and cannot be made continuous at on the projectively extended real line.