Editing Extended real number line (section)

==Motivation==
===Limits===
The extended number line is often useful to describe the behavior of a [[function (mathematics)|function]] <math>f</math> when either the [[argument of a function|argument]] <math>x</math> or the function value <math>f</math> gets "infinitely large" in some sense. For example, consider the function <math>f</math> defined by

:<math>f(x)=\frac{1}{x^{2}}</math>.

The [[graph of a function|graph]] of this function has a horizontal [[asymptote]] at <math>y=0</math>. Geometrically, when moving increasingly farther to the right along the <math>x</math>-axis, the value of <math display="inline">{1}/{x^2}</math> [[limit of a function|approaches]] 0. This limiting behavior is similar to the [[limit of a function]] <math display="inline">\lim_{x\to x_0}f(x)</math> in which the [[real number]] <math>x</math> approaches <math>x_0,</math> except that there is no real number that <math>x</math> approaches when <math>x</math> increases infinitely. Adjoining the elements <math>+\infty</math> and <math>-\infty</math> to <math>\R</math> enables a definition of "limits at infinity" which is very similar to the usual defininion of limits, except that <math>|x-x_0|<\varepsilon</math> is replaced by <math>x>N</math> (for <math>+\infty</math>) or <math>x<-N</math> (for <math>-\infty</math>). This allows proving and writing
:<math>\begin{align}\lim_{x\to+\infty}\frac1{x^2}&=0,\\\lim_{x\to-\infty}\frac1{x^2}&=0,\\\lim_{x\to0}\frac1{x^2}&=+\infty.\end{align}</math>

===Measure and integration===
{{confusing section|reason=since this is a subsection of section "Motivation", it must be understandable by readers who know nothing more than the basic definition of an integral|date=September 2024}}
In [[measure theory]], it is often useful to allow sets that have infinite [[measure (mathematics)|measure]] and integrals whose value may be infinite.

Such measures arise naturally out of calculus. For example, in assigning a measure to <math>\R</math> that agrees with the usual length of [[interval (mathematics)|intervals]], this measure must be larger than any finite real number. Also, when considering [[improper integral]]s, such as

:<math>\int_1^{\infty}\frac{dx}{x}</math>

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

:<math>f_n(x)=\begin{cases}2n(1-nx),&\mbox{if }0\leq x\leq\frac{1}{n}\\0,&\mbox{if }\frac{1}{n}<x\leq1\end{cases} </math>.

Without allowing functions to take on infinite values, such essential results as the [[monotone convergence theorem]] and the [[dominated convergence theorem]] would not make sense.