Extended real number line

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 ¯ {displaystyle {overline {mathbb {R} }}} or or ℝ ∪ {−∞, +∞}. 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 ¯ {displaystyle {overline {mathbb {R} }}} or or ℝ ∪ {−∞, +∞}. When the meaning is clear from context, the symbol +∞ is often written simply as ∞. We often wish to describe the behavior of a function f ( x ) {displaystyle f(x)} , as either the argument x {displaystyle x} or the function value f ( x ) {displaystyle f(x)} 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 x {displaystyle x} -axis, the value of 1 x 2 {displaystyle {frac {1}{x^{2}}}} 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 {displaystyle x} approaches. By adjoining the elements + ∞ {displaystyle +infty } and − ∞ {displaystyle -infty } to R {displaystyle mathbb {R} } , we allow a formulation of a 'limit at infinity' with topological properties similar to those for R {displaystyle mathbb {R} } . To make things completely formal, the Cauchy sequences definition of R {displaystyle mathbb {R} } allows us to define + ∞ {displaystyle +infty } as the set of all sequences of rationals which, for any K > 0 {displaystyle K>0} , from some point on exceed K {displaystyle K} . We can define − ∞ {displaystyle -infty } 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 R {displaystyle mathbb {R} } 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