Editing Indeterminate form (section)

== Some examples and non-examples ==
=== Indeterminate form 0/0 ===
{{Redirect|0/0|the symbol|Percent sign|0 divided by 0|Division by zero}}
<gallery>
File:Indeterminate form - x over x.gif|Fig. 1: {{var|y}} = {{sfrac|{{var|x}}|{{var|x}}}}
File:Indeterminate form - x2 over x.gif|Fig. 2: {{var|y}} = {{sfrac|{{var|x}}{{sup|2}}|{{var|x}}}}
File:Indeterminate form - sin x over x close.gif|Fig. 3: {{var|y}} = {{sfrac|sin&nbsp;{{var|x}}|{{var|x}}}}
File:Indeterminate form - complicated.gif|Fig. 4: {{var|y}} = {{sfrac|x &minus; 49|{{radic|x}} &minus; 7}} (for {{var|x}} = 49)
File:Indeterminate form - 2x over x.gif|Fig. 5: {{var|y}} = {{sfrac|{{var|a}}{{var|x}}|{{var|x}}}} where {{var|a}} = 2
File:Indeterminate form - x over x3.gif|Fig. 6: {{var|y}} = {{sfrac|{{var|x}}|{{var|x}}{{sup|3}}}}
</gallery>
The indeterminate form <math>0/0</math> is particularly common in [[calculus]], because it often arises in the evaluation of [[derivative]]s using their definition in terms of limit.

As mentioned above,
{{block indent|<math> \lim_{x \to 0} \frac{x}{x} = 1, \qquad </math> (see fig. 1)}}
while
{{block indent|<math> \lim_{x \to 0} \frac{x^{2}}{x} = 0,  \qquad </math> (see fig. 2)}}
This is enough to show that <math>0/0</math> is an indeterminate form. Other examples with this indeterminate form include
{{block indent|<math> \lim_{x \to 0} \frac{\sin(x)}{x} = 1, \qquad </math> (see fig. 3)}}
and
{{block indent|<math> \lim_{x \to 49} \frac{x - 49}{\sqrt{x}\, - 7} = 14, \qquad </math> (see fig. 4)}}
Direct substitution of the number that ''<math>x</math>'' approaches into any of these expressions shows that these are examples correspond to the indeterminate form <math>0/0</math>, but these limits can assume many different values. Any desired value <math>a</math> can be obtained for this indeterminate form as follows:
{{block indent|<math> \lim_{x \to 0} \frac{ax}{x} = a . \qquad </math> (see fig. 5)}}
The value <math>\infty</math> can also be obtained (in the sense of divergence to infinity):
{{block indent|<math> \lim_{x \to 0} \frac{x}{x^3} = \infty . \qquad </math> (see fig. 6)}}

===Indeterminate form 0<sup>0</sup> ===
{{main|Zero to the power of zero}}
{{multiple image
 | image1 = Indeterminate form - x0.gif
 | caption1 = Graph of {{math|1=''y'' = ''x''{{sup|0}}}}
 | image2 = Indeterminate form - 0x.gif
 | caption2 = Graph of {{math|1=''y'' = 0{{sup|''x''}}}}
 | 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}
 \lim_{x \to 0^+} x^0 &= 1, \\
 \lim_{x \to 0^+} 0^x &= 0.
\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
<math display="block"> \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 0<sup>0</sup> |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.

=== Expressions that are not indeterminate forms ===

The expression <math>1/0</math> is not commonly regarded as an indeterminate form, because if the limit of <math>f/g</math> exists then there is no ambiguity as to its value, as it always diverges. Specifically, if <math>f</math> approaches <math>1</math> and <math>g</math> approaches <math>0,</math> then <math>f</math> and <math>g</math> may be chosen so that:

# <math>f/g</math> approaches <math>+\infty</math>
# <math>f/g</math> approaches <math>-\infty</math>
# The limit fails to exist.

In each case the absolute value <math>|f/g|</math> approaches <math>+\infty</math>, and so the quotient <math>f/g</math> must diverge, in the sense of the [[extended real number]]s (in the framework of the [[projectively extended real line]], the limit is the [[Point at infinity|unsigned infinity]] <math>\infty</math> in all three cases<ref name=":3">{{Cite web|url=https://www.cut-the-knot.org/blue/GhostCity.shtml|title=Undefined vs Indeterminate in Mathematics|website=www.cut-the-knot.org|access-date=2019-12-02}}</ref>). Similarly, any expression of the form <math>a/0</math> with <math>a\ne0</math> (including <math>a=+\infty</math> and <math>a=-\infty</math>) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.

The expression <math>0^\infty</math> is not an indeterminate form. The expression <math>0^{+\infty}</math> obtained from considering <math>\lim_{x \to c} f(x)^{g(x)}</math> gives the limit <math>0,</math> provided that <math>f(x)</math> remains nonnegative as <math>x</math> approaches <math>c</math>. The expression <math>0^{-\infty}</math> is similarly equivalent to <math>1/0</math>; if <math>f(x) > 0</math> as <math>x</math> approaches <math>c</math>, the limit comes out as <math>+\infty</math>.

To see why, let <math>L = \lim_{x \to c} f(x)^{g(x)},</math> where <math> \lim_{x \to c} {f(x)}=0,</math> and <math> \lim_{x \to c} {g(x)}=\infty.</math> By taking the natural logarithm of both sides and using <math> \lim_{x \to c} \ln{f(x)}=-\infty,</math> we get that <math>\ln L = \lim_{x \to c} ({g(x)}\times\ln{f(x)})=\infty\times{-\infty}=-\infty,</math> which means that <math>L = {e}^{-\infty}=0.</math>