# Indeterminate form

An indeterminate form is an expression such as \( 0/0 \) or \( \infty - \infty \) that arises when evaluating a limit, in which the limiting behaviors of the parts of the expression do not by themselves determine the value of the whole. Such an expression is not a valid arithmetic statement; it signals that further analysis is needed before the limit can be found.<sup>[1](https://openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule)</sup><sup> • </sup><sup>[2](https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/01%3A_Limits/1.06%3A_Limits_Involving_Infinity)</sup>

| Key fact | Detail |
|---|---|
| Standard forms | Seven: \( 0/0 \), \( \infty/\infty \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 0^0 \), \( \infty^0 \), and \( 1^{\infty} \)<sup>[3](https://uou.ac.in/sites/default/files/slm/MT%28N%29-101.pdf)</sup> |
| What the notation means | \( 0/0 \) denotes a quotient whose numerator and denominator both shrink to 0; it is not division of zero by zero<sup>[1](https://openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule)</sup> |
| Range of outcomes | \( 0/0 \) expressions can be constructed with limit \( \infty \), 0, or any real number<sup>[2](https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/01%3A_Limits/1.06%3A_Limits_Involving_Infinity)</sup> |
| Principal tool | L'Hôpital's rule, which replaces a quotient limit of type \( 0/0 \) or \( \infty/\infty \) with the limit of the ratio of derivatives<sup>[4](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html)</sup> |
| Not indeterminate | \( 1/0 \) and \( \infty/0 \) are not indeterminate forms; the function grows without bound, so the limit is \( \infty \), \( -\infty \), or does not exist<sup>[2](https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/01%3A_Limits/1.06%3A_Limits_Involving_Infinity)</sup> |
| Origin of the rule | Published in Guillaume François Antoine, Marquis de l'Hôpital's *Analyse des infiniment petits* (1696), the first book-length calculus text, based on lectures from Johann Bernoulli<sup>[5](https://old.maa.org/press/periodicals/convergence/l-h-pital-s-rule-a-mini-primary-source-project-for-calculus-1-students)</sup> |
| In computing | In IEEE floating-point arithmetic, \( 0/0 \) raises the invalid exception and returns Not-a-Number (NaN)<sup>[6](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-02-02.html)</sup> |

## The seven standard forms

Calculus texts conventionally list seven indeterminate forms: \( 0/0 \), \( \infty/\infty \), \( 0 \cdot \infty \), \( \infty - \infty \), \( 0^0 \), \( \infty^0 \), and \( 1^{\infty} \).<sup>[3](https://uou.ac.in/sites/default/files/slm/MT%28N%29-101.pdf)</sup> Each is a shorthand for a limiting situation, not an arithmetic operation. The expression \( \infty/\infty \) means that one quantity growing without bound is divided by another quantity growing without bound; \( 0 \cdot \infty \) means that a quantity shrinking to zero is multiplied by one growing without bound. In neither case does the description alone determine what value, if any, the limit produces.<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup>

The same caution applies to the exponential forms. When \( [f(x)]^{g(x)} \) approaches \( 0^0 \), \( \infty^0 \), or \( 1^{\infty} \), the limit can take different values depending on how \( f \) and \( g \) approach their limits.<sup>[8](https://old.maa.org/node/116809)</sup> A rigorous analysis treats these cases as indeterminate expressions even for the simplest functions, sequences: knowing only the limits of the parts never suffices, because the answer depends on the properties of the particular functions involved.<sup>[9](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.04%3A_Infinite_Limits._Operations_in_E)</sup>

## Why the forms are indeterminate

The label "indeterminate" reflects the fact that the same symbol can lead to different limits. The limits \( \lim_{x \to 0} \frac{\sin x}{x} \) and \( \lim_{x \to 1} \frac{x^2-1}{x-1} \) both return the form \( 0/0 \) when the limiting value is substituted, yet their values are 1 and 2 respectively. With suitable constructions, \( 0/0 \) expressions can be made to have a limit of \( \infty \), 0, or any other real number.<sup>[2](https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/01%3A_Limits/1.06%3A_Limits_Involving_Infinity)</sup>

What actually happens in a \( 0/0 \) limit is a race of rates: the numerator shrinks to 0 while the denominator also shrinks to 0, and the relative speeds determine the outcome. Similarly, \( \infty - \infty \) describes two quantities both growing without bound, and their difference can be made to take any value between \( -\infty \) and \( \infty \).<sup>[2](https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/01%3A_Limits/1.06%3A_Limits_Involving_Infinity)</sup> Concrete sequence examples show the spread: with \( u_m = 2m \) and \( v_m = -m \), the sum \( u_m + v_m \to +\infty \); with \( x_m = 2m \) and \( y_m = -2m \), the sum is constant with limit 0.<sup>[9](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.04%3A_Infinite_Limits._Operations_in_E)</sup>

## L'Hôpital's rule

The principal tool for resolving quotient indeterminate forms is [L'Hôpital's rule](https://www.edgepedia.org/lhopitals-rule). In its basic form, suppose \( \lim_{x \to a} f(x) = 0 \) and \( \lim_{x \to a} g(x) = 0 \), with \( f \) and \( g \) differentiable on an open interval containing \( a \) and \( g'(x) \neq 0 \) there except possibly at \( a \) itself. Then

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \]

provided the limit on the right exists, whether finite, \( +\infty \), or \( -\infty \). A parallel version covers the case where both functions tend to \( \pm\infty \), and similar statements hold as \( x \to \pm\infty \).<sup>[4](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html)</sup> The numerator and denominator are differentiated separately; the rule is not the quotient rule for derivatives.<sup>[4](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html)</sup>

The intuition behind the \( 0/0 \) case comes from local linear approximations: near \( a \), \( f(x) \approx f(a) + f'(a)(x-a) \) and \( g(x) \approx g(a) + g'(a)(x-a) \), so the ratio \( f(x)/g(x) \) behaves like the ratio of the derivatives when both function values vanish at \( a \).<sup>[1](https://openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule)</sup>

**Worked examples.** The rule resolves \( \lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1 \), and repeated application handles \( \lim_{x \to 0} \frac{x^2}{1 - \cos x} = \lim_{x \to 0} \frac{2x}{\sin x} = \lim_{x \to 0} \frac{2}{\cos x} = 2 \).<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup> The rule also compares growth rates: \( \lim_{x \to \infty} \frac{e^x}{x^3} = \infty \) after three applications, showing that the exponential outgrows the cubic, while \( \lim_{x \to \infty} \frac{\log x}{x} = \lim_{x \to \infty} \frac{1/x}{1} = 0 \).<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup>

## Reducing the other forms

L'Hôpital's rule applies only to ratios, so the five non-quotient forms are first rewritten algebraically as \( 0/0 \) or \( \infty/\infty \).<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup>

- **Products** \( 0 \cdot \infty \): flip one factor. For example, \( \lim_{x \to 0^+} \sqrt{x} \ln x \) is rewritten as \( \frac{\ln x}{1/\sqrt{x}} \), a \( -\infty/\infty \) quotient, and evaluates to 0.<sup>[4](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html)</sup>
- **Differences** \( \infty - \infty \): combine over a common denominator. The limit \( \lim_{x \to (\pi/2)^-} (\sec x - \tan x) \) becomes \( \frac{1 - \sin x}{\cos x} \), a \( 0/0 \) form with value 0.<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup>
- **Powers** \( 0^0 \), \( \infty^0 \), \( 1^{\infty} \): write \( z = e^{\ln z} \) and evaluate the exponent's limit. For \( \lim_{x \to 0^+} x^x \), the exponent \( x \ln x \) is a \( 0 \cdot (-\infty) \) form that reduces to a quotient and gives \( e^0 = 1 \).<sup>[4](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html)</sup> The same logarithm technique handles \( \lim_{x \to 0} (1+x)^{a/x} \), a \( 1^{\infty} \) form that arises naturally in systems with exponential growth or decay.<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup>

## Cautions and failures

The rule's hypotheses must be checked before each use. Applying it to a limit that is not indeterminate produces wrong answers: \( \lim_{x \to -3} \frac{x^3+27}{x^2+9} = \frac{0}{18} = 0 \), but inappropriate use of the rule would give the incorrect value \( -9/2 \).<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup> The rule also requires that the limit of the ratio of derivatives exist; when it does not, the rule says nothing, and other methods are needed. Robert Boas, an American mathematician, published "Counterexamples to L'Hopital's rule" in the *American Mathematical Monthly* in 1986, collecting cases where the rule misleads.<sup>[10](https://mpra.ub.uni-muenchen.de/79069/)</sup> Naive repeated application can also cycle indefinitely: for \( \lim_{x \to \infty} \frac{e^x + e^{-x}}{e^x - e^{-x}} \), each differentiation returns the same \( \infty/\infty \) form, and the correct approach is to factor out \( e^x \) instead.<sup>[7](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)</sup>

## History

The rule carries the name of [Guillaume François Antoine, Marquis de l'Hôpital](https://www.edgepedia.org/guillaume-de-lhopital) (1661–1704), a French nobleman who engaged [Johann Bernoulli](https://www.edgepedia.org/johann-bernoulli) to teach him the new calculus of Leibniz and the Bernoulli brothers in Paris in 1691. L'Hôpital kept detailed notes of the lectures, and from them emerged *Analyse des infiniment petits pour l'intelligence des lignes courbes*, published in 1696, the first book-length account of differential calculus. Its best-known result is the theorem for limits of indeterminate type \( 0/0 \), stated in terms of differentials of numerator and denominator.<sup>[5](https://old.maa.org/press/periodicals/convergence/l-h-pital-s-rule-a-mini-primary-source-project-for-calculus-1-students)</sup> The book brought Leibniz's differential notation into general use in continental Europe.<sup>[11](https://library.brown.edu/exhibits/archive/math/nofr.html)</sup> The arrangement behind the book was unusual: in a letter of 17 March 1694, l'Hôpital proposed paying Bernoulli for his discoveries, an agreement that explains why results from Bernoulli's lectures appear under l'Hôpital's name.<sup>[12](https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bernoulli/index.html)</sup>

The English terminology is more recent. "Indeterminate Forms" appears as a chapter title in 1841 in *An Elementary Treatise on Curves, Functions, and Forces* by Benjamin Pierce, and forms such as \( 0/0 \) were called "singular values and singular forms" in 1849 in the second edition of James Thomson's *An Introduction to the Differential and Integral Calculus*.<sup>[3](https://uou.ac.in/sites/default/files/slm/MT%28N%29-101.pdf)</sup>

## Indeterminate forms in computing

In IEEE floating-point arithmetic the same expression behaves differently from the limit notion. An operation such as \( 0/0 \) raises the invalid exception and returns Not-a-Number (NaN), while \( 1/0 \) raises the divide-by-zero exception and returns an infinite value; by default these exceptions are flagged rather than stopping the computation.<sup>[6](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-02-02.html)</sup> The distinction matters in practice: a limit that is analytically a \( 0/0 \) indeterminate form with a finite value, such as a difference quotient as the step size approaches zero, will produce NaN in floating point if evaluated by direct substitution, so numerical code must handle the limiting case separately.<sup>[6](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-02-02.html)</sup>

## References

1. [L'Hôpital's Rule, Calculus Volume 1, OpenStax](https://openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule)
2. [Limits Involving Infinity, Calculus 3e (Apex), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_%28Apex%29/01%3A_Limits/1.06%3A_Limits_Involving_Infinity)
3. [Calculus MT(N) 101, Unit 5: Indeterminate Forms, Uttarakhand Open University](https://uou.ac.in/sites/default/files/slm/MT%28N%29-101.pdf)
4. [L'Hopital's Rule, University of California, Davis (Duane Kouba)](https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/lhopitaldirectory/LHopital.html)
5. [L'Hôpital's Rule: A Mini-Primary Source Project for Calculus 1 Students, Convergence, Mathematical Association of America](https://old.maa.org/press/periodicals/convergence/l-h-pital-s-rule-a-mini-primary-source-project-for-calculus-1-students)
6. [CS 4220: Numerical Analysis, Cornell University](https://www.cs.cornell.edu/courses/cs4220/2026sp/lec/2026-02-02.html)
7. [L'Hôpital's Rule, APEX Calculus, University of North Dakota](https://sites.und.edu/timothy.prescott/apex/web/apex.Ch7.S5.html)
8. [What is 0^0? Indeterminate Forms, Mathematical Association of America](https://old.maa.org/node/116809)
9. [Infinite Limits. Operations in E*, Mathematical Analysis (Zakon), Mathematics LibreTexts](https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_%28Zakon%29/04%3A_Function_Limits_and_Continuity/4.04%3A_Infinite_Limits._Operations_in_E)
10. [Abueg, Luisito (2016). A historical walkthrough with L'Hospital, MPRA Paper 79069](https://mpra.ub.uni-muenchen.de/79069/)
11. [From Euclid to Newton, Brown University Library exhibition](https://library.brown.edu/exhibits/archive/math/nofr.html)
12. [Bernoulli and l'Hôpital, Math 1a exhibit, Harvard University](https://people.math.harvard.edu/~knill/teaching/math1a_2011/exhibits/bernoulli/index.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Differential calculus and derivatives*

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