section_path: Real exponents/Limits of rational exponents
section_name: Limits of rational exponents
paragraph_text: Limits of rational exponents
  
  Since any irrational number can be expressed as the limit of a sequence of rational numbers, exponentiation of a positive real number b with an arbitrary real exponent x can be defined by continuity with the rule
  
  where the limit is taken over rational values of r only. This limit exists for every positive b and every real x.
  
  For example, if x = π, the non-terminating decimal representation π = 3.14159... and the monotonicity of the rational powers can be used to obtain intervals bounded by rational powers that are as small as desired, and must contain b π: {\displaystyle b^{\pi }:}
  
  So, the upper bounds and the lower bounds of the intervals form two sequences that have the same limit, denoted b π. {\displaystyle b^{\pi }.}
  
  This defines b x {\displaystyle b^{x}} for every positive b and real x as a continuous function of b and x. See also Well-defined expression.