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{{shortShort description|TermExpression in mathematical analysis}}

In [[calculus]], it is usually possible to compute the [[limit (mathematics)|limit]] of the sum, difference, product, quotient or power of two functions by taking the corresponding combination of the separate limits of each respective function. For example,

In [[calculus]] and other branches of [[mathematical analysis]], limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their [[limit (mathematics)|limits]]; if the expression obtained after this substitution does not provide sufficient information to determine the original limit, then the expression is called an '''indeterminate form'''. More specifically, an indeterminate form is a mathematical expression involving at most two of <math>0~</math>, <math>1</math> or <math>\infty</math>, obtained by applying the [[algebraic limit theorem]] in the process of attempting to determine a limit, which fails to restrict that limit to one specific value or infinity, and thus does not determine the limit being sought. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity).<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Indeterminate.html|title=Indeterminate|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}</ref> The term was originally introduced by [[Cauchy]]'s student [[Moigno]] in the middle of the 19th century.

<math display=block>\begin{align}

There are seven indeterminate forms which are typically considered in the literature:<ref name=":1" />

\lim_{x \to c} \bigl(f(x) + g(x)\bigr) &= \lim_{x \to c} f(x) + \lim_{x \to c} g(x), \\[3mu]

\lim_{x \to c} \bigl(f(x)g(x)\bigr) &= \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x),

\end{align}</math>

and likewise for other arithmetic operations; this is sometimes called the [[limit of a function#Properties|algebraic limit theorem]]. However, certain combinations of particular limiting values cannot be computed in this way, and knowing the limit of each function separately does not suffice to determine the limit of the combination. In these particular situations, the limit is said to take an '''indeterminate form''', described by one of the informal expressions

:<math>\frac 00,~ \frac{\infty}{\infty},~ 0\times\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text{ and } \infty^0 .</math>

:<math display=block>\frac 00,~ \frac{\infty}{\infty},~ 0\times\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text{ andor } \infty^0 .,</math>

The most common example of an indeterminate form occurs when determining the limit of the ratio of two functions, in which both of these functions tend to zero in the limit, and is referred to as "the indeterminate form <math>0/0</math>". For example, as <math>x</math> approaches <math>0~</math>, the ratios <math>x/x^3</math>, <math>x/x</math>, and <math>x^2/x</math> go to <math>\infty</math>, <math>1</math>, and <math>0~</math> respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is <math>0/0</math>, which is undefined. In a loose manner of speaking, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, and it is easy to construct similar examples for which the limit is any particular value.

among a wide variety of uncommon others, where each expression stands for the limit of a function constructed by an arithmetical combination of two functions whose limits respectively tend to {{tmath|0,}} {{tmath|1,}} or {{tmath|\infty}} as indicated.{{sfnp|Varberg|Purcell|Rigdon|2007|p=423, 429, 430, 431, 432}}

So, given that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> both approaching <math>0~</math> as <math>x</math> approaches some [[limit point]] <math>c</math>, that fact alone does not give enough information for evaluating the [[limit of a function|limit]]

A limit taking one of these indeterminate forms might tend to zero, might tend to any finite value, might tend to infinity, or might diverge, depending on the specific functions involved. A limit which unambiguously tends to infinity, for instance <math display=inline>\lim_{x \to 0} 1/x^2 = \infty,</math> is not considered indeterminate.<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Indeterminate.html|title=Indeterminate|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}</ref> The term was originally introduced by [[Cauchy]]'s student [[Moigno]] in the middle of the 19th century.

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

The most common example of an indeterminate form occurs when determiningis the limit of the ratioquotient of two functions, in which botheach of thesewhich functions tendconverges to zero. inThis theindeterminate limit, andform is referreddenoted to as "the indeterminate formby <math>0/0</math>". For example, as <math>x</math> approaches <math>0~,</math>, the ratios <math>x/x^3</math>, <math>x/x</math>, and <math>x^2/x</math> go to <math>\infty</math>, <math>1</math>, and <math>0~</math> respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is <math>0/0</math>, which is undefinedindeterminate. In a loose manner ofthis speakingsense, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, andby itappropriate ischoices easyof functions to constructput similarin examplesthe numerator and denominator. A pair of functions for which the limit is any particular given value may in fact be found. Even more surprising, perhaps, the quotient of the two functions may in fact diverge, and not merely diverge to infinity. For example, <math> x \sin(1/x) / x</math>.

Not every undefined algebraic expression corresponds to an indeterminate form.<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> For example, the expression <math>1/0</math> is undefined as a [[real number]] but does not correspond to an indeterminate form; any defined limit that gives rise to this form will diverge to infinity.

So, giventhe fact that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> bothconverge approachingto <math>0~</math> as <math>x</math> approaches some [[limit point]] <math>c</math>, that fact alone does not give enoughis informationinsufficient forto evaluatingdeterminate the [[limit of a function|limit]]

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

An expression that arises by ways other than applying the algebraic limit theorem may have the same form of an indeterminate form. However it is not appropriate to call an expression "indeterminate form" if the expression is made outside the context of determining limits.

AnotherAn example is the expression <math>0^0</math>. Whether this expression is left undefined, or is defined to equal <math>1</math>, depends on the field of application and may vary between authors. For more, see the article [[Zero to the power of zero]]. Note that <math>0^\infty</math> and other expressions involving infinity [[#Expressions that are not indeterminate forms|are not indeterminate forms]].

For example, <math>0/0</math> which arises from substituting <math>0~</math> for <math>x</math> in the equation <math>f(x)=|x|/(|x-1|-1)</math> is not an indeterminate form since this expression is not made in the determination of a limit (it is in fact undefined as [[division by zero]]).

Another example is the expression <math>0^0</math>. Whether this expression is left undefined, or is defined to equal <math>1</math>, depends on the field of application and may vary between authors. For more, see the article [[Zero to the power of zero]]. Note that <math>0^\infty</math> and other expressions involving infinity [[#Expressions that are not indeterminate forms|are not indeterminate forms]].

== 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}}}}

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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}

\lim_{x \to 0^+} x^0 &= 1, \\

\lim_{x \to 0^+} 0^x &= 0.

\end{align} </math>

{{blockThus, in general, knowing that indent|<math> \textstyle\lim_{x \to 0^+c} f(x^0) \;=\; 10</math> ,and <math>\qquadtextstyle\lim_{x \to c} g(x) \;=\; 0</math> (seeis fig.not 7)}}sufficient to evaluate the limit

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

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>\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.

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=== 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>

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# 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>

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Suppose there are two equivalent infinitesimals <math>\alpha \sim \alpha'</math> and <math>\beta \sim \beta'</math>.

:<math display=block>\lim \frac{\beta}{\alpha} = \lim \frac{\beta \beta' \alpha'}{\beta' \alpha' \alpha} = \lim \frac{\beta}{\beta'} \lim \frac{\alpha'}{\alpha} \lim \frac{\beta'}{\alpha'} = \lim \frac{\beta'}{\alpha'}</math>

For the evaluation of the indeterminate form <math>0/0</math>, one can make use of the following facts about equivalent [[infinitesimal]]s (e.g., <math>x\sim\sin x</math> if ''x'' becomes closer to zero):<ref>{{Cite web|url=http://www.vaxasoftware.com/doc_eduen/mat/infiequi.pdf|title=Table of equivalent infinitesimals|website=Vaxa Software}}</ref>

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{{block indent|<math>e^x - 1\sim x,</math>}}

{{block indent|<math>(1 + x)^a - 1 \sim ax.</math>}}

For example:

:<math>\begin{align}\lim_{x \to 0} \frac{1}{x^3} \left[\left(\frac{2+\cos x}{3}\right)^x - 1 \right] & = \lim_{x \to 0} \frac{e^{x\ln{\frac{2 + \cos x}{3}}}-1}{x^3} \\ & = \lim_{x \to 0} \frac{1}{x^2} \ln \frac{2+ \cos x}{3} \\ & = \lim_{x \to 0} \frac{1}{x^2} \ln \left(\frac{\cos x -1}{3}+1\right) \\ &= \lim_{x \to 0} \frac{\cos x -1}{3x^2} \\ &= \lim_{x \to 0} -\frac{x^2}{6x^2} \\ & = -\frac{1}{6}\end{align}</math>

<math display=block>\begin{align}

\lim_{x \to 0} \frac{1}{x^3} \left[\left(\frac{2+\cos x}{3}\right)^x - 1 \right]

&= \lim_{x \to 0} \frac{e^{x\ln{\frac{2 + \cos x}{3}}}-1}{x^3} \\

&= \lim_{x \to 0} \frac{1}{x^2} \ln \frac{2+ \cos x}{3} \\

&= \lim_{x \to 0} \frac{1}{x^2} \ln \left(\frac{\cos x -1}{3}+1\right) \\

&= \lim_{x \to 0} \frac{\cos x -1}{3x^2} \\

&= \lim_{x \to 0} -\frac{x^2}{6x^2} \\

&= -\frac{1}{6}

\end{align}</math>

In the 2nd equality, <math>e^y - 1 \sim y</math> where <math>y = x\ln{2+\cos x \over 3}</math> as ''y'' become closer to 0 is used, and <math>y \sim \ln {(1+y)}</math> where <math>y = {{\cos x - 1} \over 3}</math> is used in the 4th equality, and <math>1-\cos x \sim {x^2 \over 2}</math> is used in the 5th equality.

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== List of indeterminate forms ==

The following table lists the most common indeterminate forms, and the transformations for applying l'Hôpital's rule.

{| border=1 class="wikitable" style="background-color: #ffffff; width: 85%;"

!Indeterminate form

!Conditions

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!Transformation to <math>\infty/\infty</math>

|-

|{{sfrac|<math>0</math>|<math>0</math>}}

|<math> \lim_{x \to c} f(x) = 0,\ \lim_{x \to c} g(x) = 0 \! </math>

|<{{center>|&mdash;</center>}}

|<math> \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{1/g(x)}{1/f(x)} \! </math>

|-

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|<math> \lim_{x \to c} f(x) = \infty,\ \lim_{x \to c} g(x) = \infty \! </math>

|<math> \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{1/g(x)}{1/f(x)} \! </math>

|<{{center>|&mdash;</center>}}

|-

|<math>0\cdot\infty</math>

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== References ==

=== Citations ===

{{reflist}}

=== Bibliographies ===

* {{cite book

| last1 = Varberg | first1 = Dale E.

| last2 = Purcell | first2 = Edwin J.

| last3 = Rigdon | first3 = Steven E.

| title = Calculus

| year = 2007

| publisher = [[Pearson Prentice Hall]]

| edition = 9th

| isbn = 978-0131469686

}}

{{Calculus topics}}