Indeterminate form: Difference between revisions

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{{Short description|Expression in mathematical analysis}}

'''Indeterminate form''' is a mathematical expression that can obtain any value depending on circumstances. In [[calculus]], itand isother usuallybranches possibleof to[[mathematical analysis]], computewhen the [[limit (mathematics)|limit]] of the sum, difference, product, quotient or power of two functions byis takingtaken, it may often be possible to simply add, subtract, multiply, divide or exponentiate the corresponding combinationlimits of these two functions respectively. However, there are occasions where it is unclear what the separatesum, limitsdifference, product, quotient, or power of eachthese respectivetwo functionlimits ought to be. For example, it is unclear what the following expressions ought to evaluate to:<ref name=":1" />

* <math>0 \div 0</math>

<math display=block>\begin{align}

* <math> {\infty} \div {\infty} </math>

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

* <math> 0\times\infty </math>

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

* <math> \end{align}infty - \infty </math>

* <math> 0^0 </math>

* <math> 1^\infty </math>

* <math> \infty^0 </math>

* <math> \sqrt[0]{1} </math>

* <math> \sqrt[\infty]{0} </math>

* <math> \sqrt[\infty]{\infty} </math>

* {{math|log₀(0)}}

* {{math|log₁(1)}}

* {{math|log_∞(∞)}}

AThese limitseven takingexpressions oneare ofknown theseas '''indeterminate forms'''. mightMore tendspecifically, such expressions are obtained by naively applying the [[algebraic limit theorem]] to zeroevaluate the limit of the corresponding arithmetic operation of two functions, mightyet tendthere are examples of pairs of functions that after being operated on converge to any0, converge to another finite value, might tenddiverge to infinity, or mightjust diverge,. dependingThis oninability to decide what the specificlimit functionsought involvedto be explains why these forms are regarded as '''indeterminate'''. A limit which unambiguously tendsconfirmed to be infinity, foris instancenot <mathindeterminate display=inline>\lim_{xsince \toit 0}has 1/x^2been =determined \infty,</math>to ishave nota consideredspecific indeterminatevalue (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.

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

The most common example of an indeterminate form is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by <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 indeterminate. In this sense, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, by appropriate choices of functions to put in the 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>.

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

So the fact that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> converge to <math>0~</math> as <math>x</math> approaches some [[limit point]] <math>c</math> is insufficient to determinate the [[limit of a function|limit]]

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}}

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.

The most common example of an indeterminate form is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by <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 indeterminate. In this sense, <math>0/0</math> can take on the values <math>0</math>, <math>1</math>, or <math>\infty</math>, by appropriate choices of functions to put in the 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>.

So the fact that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> converge to <math>0</math> as <math>x</math> approaches some [[limit point]] <math>c</math> is insufficient to determinate 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.

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]]).

AnAnother 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 ==

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===Indeterminate form 0⁰ ===

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

{{multiple image

<gallery>

| image1 = Indeterminate form - x0.gif

|File:Indeterminate caption1form =- Graphx0.gif|Fig. of7: {{mathvar|1=''y''}} = ''{{var|x''}}{{sup|0}}}}

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

</gallery>

| 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}

{{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>}}

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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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>

<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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== 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}}