Editing Exponentiation (section)

===Limits of rational exponents===
[[File:Continuity of the Exponential at 0.svg|thumb|The limit of {{math|''e''{{sup|1/''n''}}}} is {{math|1=''e''{{sup|0}} = 1}} when {{mvar|n}} tends to the infinity.]]
Since any [[irrational number]] can be expressed as the [[limit of a sequence]] of rational numbers, exponentiation of a positive real number {{mvar|b}} with an arbitrary real exponent {{mvar|x}} can be defined by [[continuous function|continuity]] with the rule<ref name="Denlinger">{{cite book |title=Elements of Real Analysis |last=Denlinger |first=Charles G. |publisher=Jones and Bartlett |date=2011 |pages=278–283 |isbn=978-0-7637-7947-4}}</ref>
:<math> b^x = \lim_{r (\in \mathbb{Q}) \to x} b^r \quad (b \in \mathbb{R}^+,\, x \in \mathbb{R}),</math>
where the limit is taken over rational values of {{mvar|r}} only. This limit exists for every positive {{mvar|b}} and every real {{mvar|x}}.

For example, if {{math|1=''x'' = {{pi}}}}, the [[non-terminating decimal]] representation {{math|1=''π'' = 3.14159...}} and the [[monotone function|monotonicity]] of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain <math>b^\pi:</math>
:<math>\left[b^3, b^4\right], \left[b^{3.1}, b^{3.2}\right], \left[b^{3.14}, b^{3.15}\right], \left[b^{3.141}, b^{3.142}\right], \left[b^{3.1415}, b^{3.1416}\right], \left[b^{3.14159}, b^{3.14160}\right], \ldots</math>
So, the upper bounds and the lower bounds of the intervals form two [[sequence (mathematics)|sequences]] that have the same limit, denoted <math>b^\pi.</math>

This defines <math>b^x</math> for every positive {{mvar|b}} and real {{mvar|x}} as a [[continuous function]] of {{mvar|b}} and {{mvar|x}}. See also ''[[Well-defined expression]]''.<ref>{{cite book |chapter=Limits of sequences |chapter-url={{Google books |ecTsDAAAQBAJ |page=154 |plainurl=yes}} |title=Analysis I |series=Texts and Readings in Mathematics |year=2016 |last1=Tao |first1=Terence |volume=37 |pages=126–154 |isbn=978-981-10-1789-6 |doi=10.1007/978-981-10-1789-6_6}}</ref>