Editing Exponentiation (section)

==Limits of powers==
[[Zero to the power of zero]] gives a number of examples of limits that are of the [[indeterminate form]] 0<sup>0</sup>. The limits in these examples exist, but have different values, showing that the two-variable function {{math|''x''<sup>''y''</sup>}} has no limit at the point {{math|(0, 0)}}. One may consider at what points this function does have a limit.

More precisely, consider the function <math>f(x,y) = x^y</math> defined on <math> D = \{(x, y) \in \mathbf{R}^2 : x > 0 \}</math>. Then {{math|''D''}} can be viewed as a subset of {{math|{{overline|'''R'''}}<sup>2</sup>}} (that is, the set of all pairs {{math|(''x'', ''y'')}} with {{math|''x''}}, {{math|''y''}} belonging to the [[extended real number line]] {{math|1={{overline|'''R'''}} = [−∞, +∞]}}, endowed with the [[product topology]]), which will contain the points at which the function {{math|''f''}} has a limit.

In fact, {{math|''f''}} has a limit at all [[accumulation point]]s of {{math|''D''}}, except for {{math|(0, 0)}}, {{math|(+∞, 0)}}, {{math|(1, +∞)}} and {{math|(1, −∞)}}.<ref>Nicolas Bourbaki, ''Topologie générale'', V.4.2.</ref> Accordingly, this allows one to define the powers {{math|''x''<sup>''y''</sup>}} by continuity whenever {{math|0 ≤ ''x'' ≤ +∞}}, {{math|−∞ ≤ y ≤ +∞}}, except for {{math|0<sup>0</sup>}}, {{math|(+∞)<sup>0</sup>}}, {{math|1<sup>+∞</sup>}} and {{math|1<sup>−∞</sup>}}, which remain indeterminate forms.

Under this definition by continuity, we obtain:
* {{math|1=''x''<sup>+∞</sup> = +∞}} and {{math|1=''x''<sup>−∞</sup> = 0}}, when {{math|1 < ''x'' ≤ +∞}}.
* {{math|1=''x''<sup>+∞</sup> = 0}} and {{math|1=''x''<sup>−∞</sup> = +∞}}, when {{math|0 < ''x'' < 1}}.
* {{math|1=0<sup>''y''</sup> = 0}} and {{math|1=(+∞)<sup>''y''</sup> = +∞}}, when {{math|0 < ''y'' ≤ +∞}}.
* {{math|1=0<sup>''y''</sup> = +∞}} and {{math|1=(+∞)<sup>''y''</sup> = 0}}, when {{math|−∞ ≤ ''y'' < 0}}.

These powers are obtained by taking limits of {{math|''x''<sup>''y''</sup>}} for ''positive'' values of {{math|''x''}}. This method does not permit a definition of {{math|''x''<sup>''y''</sup>}} when {{math|''x'' < 0}}, since pairs {{math|(''x'', ''y'')}} with {{math|''x'' < 0}} are not accumulation points of {{math|''D''}}.

On the other hand, when {{math|''n''}} is an integer, the power {{math|''x''<sup>''n''</sup>}} is already meaningful for all values of {{math|''x''}}, including negative ones. This may make the definition {{math|1=0<sup>''n''</sup> = +∞}} obtained above for negative {{math|''n''}} problematic when {{math|''n''}} is odd, since in this case {{math|''x''<sup>''n''</sup> → +∞}} as {{math|''x''}} tends to {{math|0}} through positive values, but not negative ones.