Indeterminate form: Difference between revisions

Revision as of 00:55, 10 March 2024

Latest revision as of 19:07, 3 July 2025

(20 intermediate revisions by 10 users not shown)

Line 2:

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,

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

\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<ref name=":1" />

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

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}}

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.

Line 26:

== 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}}}}

Line 52:

===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.

Line 95 ⟶ 98:

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>

Line 110 ⟶ 114:

{{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.

Line 130 ⟶ 145:

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

Line 182 ⟶ 197:

== 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}}