Indeterminate form: Difference between revisions

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===Indeterminate form 0⁰ ===

{{main|Zero to the power of zero}}

{{multiple image

| image1 = Indeterminate form - x0.gif

<gallery>

File:Indeterminate | formcaption1 -= x0.gif|Fig.Graph 7:of {{varmath|1=''y}}'' = {{var|''x}}''{{sup|0}}}}

File: | image2 = Indeterminate form - 0x.gif|Fig. 8: {{var|y}} = 0{{sup|{{var|x}}}}

| caption2 = Graph of {{math|1=''y'' = 0{{sup|''x''}}}}

</gallery>

| total_width = 300

| direction = vertical

}}

The following limits illustrate that the expression <math>0^0</math> is an indeterminate form:

<math display="block"> \begin{align}

{{block indent|<math> \lim_{x \to 0^+} x^0 &= 1 , \qquad </math> (see fig. 7)}}\

{{block indent|<math> \lim_{x \to 0^+} 0^x &= 0 . \qquad </math> (see fig. 8)}}

\end{align} </math>

Thus, in general, knowing that <math>\textstyle\lim_{x \to c} f(x) \;=\; 0</math> and <math>\textstyle\lim_{x \to c} g(x) \;=\; 0</math> is not sufficient to evaluate the limit

{{block indent|<math display="block"> \lim_{x \to c} f(x)^{g(x)}. .</math>}}

{{block indent|<math>\lim_{x \to c} f(x)^{g(x)} .</math>}}

If the functions <math>f</math> and <math>g</math> are [[Analytic function|analytic]] at <math>c</math>, and <math>f</math> is positive for <math>x</math> sufficiently close (but not equal) to <math>c</math>, then the limit of <math>f(x)^{g(x)}</math> will be <math>1</math>.<ref>{{cite journal |doi=10.2307/2689754 |author1=Louis M. Rotando |author2=Henry Korn |title=The indeterminate form 0⁰ |journal=Mathematics Magazine |date=January 1977 |volume=50 |issue=1 |pages=41&ndash;42|jstor=2689754 }}</ref> Otherwise, use the transformation in the [[#List of indeterminate forms|table]] below to evaluate the limit.