Indeterminate form

Last updated

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 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 , and , 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 (if a limit is confirmed as infinity, then it is not indeterminate since the limit is determined as infinity) and thus does not yet determine the limit being sought. [1] [2] The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.


There are seven indeterminate forms which are typically considered in the literature: [2]

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 ". For example, as approaches , the ratios , , and go to , , and respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is , which is undefined. In a loose manner of speaking, can take on the values , , or , and it is easy to construct similar examples for which the limit is any particular value.

So, given that two functions and both approaching as approaches some limit point , that fact alone does not give enough information for evaluating the limit

Not every undefined algebraic expression corresponds to an indeterminate form. [3] For example, the expression 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.

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, which arises from substituting for in the equation 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 . Whether this expression is left undefined, or is defined to equal , depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that and other expressions involving infinity are not indeterminate forms.

Some examples and non-examples

Indeterminate form 0/0

The indeterminate form is particularly common in calculus, because it often arises in the evaluation of derivatives using their definition in terms of limit.

As mentioned above,

(see fig. 1)


(see fig. 2)

This is enough to show that is an indeterminate form. Other examples with this indeterminate form include

(see fig. 3)


(see fig. 4)

Direct substitution of the number that approaches into any of these expressions shows that these are examples correspond to the indeterminate form , but these limits can assume many different values. Any desired value can be obtained for this indeterminate form as follows:

(see fig. 5)

The value can also be obtained (in the sense of divergence to infinity):

(see fig. 6)

Indeterminate form 00

The following limits illustrate that the expression is an indeterminate form:

(see fig. 7)
(see fig. 8)

Thus, in general, knowing that and is not sufficient to evaluate the limit

If the functions and are analytic at , and is positive for sufficiently close (but not equal) to , then the limit of will be . [4] Otherwise, use the transformation in the table below to evaluate the limit.

Expressions that are not indeterminate forms

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

  1. approaches
  2. approaches
  3. The limit fails to exist.

In each case the absolute value approaches , and so the quotient must diverge, in the sense of the extended real numbers (in the framework of the projectively extended real line, the limit is the unsigned infinity in all three cases [3] ). Similarly, any expression of the form with (including and ) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.

The expression is not an indeterminate form. The expression obtained from considering gives the limit , provided that remains nonnegative as approaches . The expression is similarly equivalent to ; if as approaches , the limit comes out as .

To see why, let where and By taking the natural logarithm of both sides and using we get that which means that

Evaluating indeterminate forms

The adjective indeterminate does not imply that the limit does not exist, as many of the examples above show. In many cases, algebraic elimination, L'Hôpital's rule, or other methods can be used to manipulate the expression so that the limit can be evaluated. [1]

Equivalent infinitesimal

When two variables and converge to zero at the same limit point and , they are called equivalent infinitesimal (equiv. ).

Moreover, if variables and are such that and , then:

Here is a brief proof:

Suppose there are two equivalent infinitesimals and .

For the evaluation of the indeterminate form , one can make use of the following facts about equivalent infinitesimals (e.g., if x becomes closer to zero): [5]

For example:

In the 2nd equality, where as y become closer to 0 is used, and where is used in the 4th equality, and is used in the 5th equality.

L'Hôpital's rule

L'Hôpital's rule is a general method for evaluating the indeterminate forms and . This rule states that (under appropriate conditions)

where and are the derivatives of and . (Note that this rule does not apply to expressions , , and so on, as these expressions are not indeterminate forms.) These derivatives will allow one to perform algebraic simplification and eventually evaluate the limit.

L'Hôpital's rule can also be applied to other indeterminate forms, using first an appropriate algebraic transformation. For example, to evaluate the form 00:

The right-hand side is of the form , so L'Hôpital's rule applies to it. Note that this equation is valid (as long as the right-hand side is defined) because the natural logarithm (ln) is a continuous function; it is irrelevant how well-behaved and may (or may not) be as long as is asymptotically positive. (the domain of logarithms is the set of all positive real numbers.)

Although L'Hôpital's rule applies to both and , one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards). One can change between these forms, if necessary, by transforming to .

List of indeterminate forms

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

Indeterminate formConditionsTransformation to Transformation to

See also

Related Research Articles

LHôpitals rule Mathematical rule for evaluating certain limits

In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule provides a technique to evaluate limits of indeterminate forms. Application of the rule often converts an indeterminate form to an expression that can be easily evaluated by substitution. The rule is named after the 17th-century French mathematician Guillaume de l'Hôpital. Although the rule is often attributed to L'Hôpital, the theorem was first introduced to him in 1694 by the Swiss mathematician Johann Bernoulli.

In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable to a function of a complex variable . The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication.

Exponential distribution Probability distribution

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

Beta distribution Probability distribution

In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by α and β, that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet distribution.

Gamma distribution Probability distribution

In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. There are two different parameterizations in common use:

  1. With a shape parameter k and a scale parameter θ.
  2. With a shape parameter α = k and an inverse scale parameter β = 1/θ, called a rate parameter.

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

Beta function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral

In quantum mechanics, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic crystal lattice. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice. It is a generalization of the free electron model, which assumes zero potential inside the lattice.

Stable distribution Distribution of variables which satisfies a stability property under linear combinations

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

Inverse-gamma distribution Two-parameter family of continuous probability distributions

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required.

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.


A biarc is a smooth curve formed from two circular arcs. In order to make the biarc smooth, the two arcs should have the same tangent at the connecting point where they meet.

In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.

In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that uses the series solution for a differential equation, where we assume the solution takes the form of a series. This is usually the method we use for complicated ordinary differential equations.

Flamant solution

The Flamant solution provides expressions for the stresses and displacements in a linear elastic wedge loaded by point forces at its sharp end. This solution was developed by A. Flamant in 1892 by modifying the three-dimensional solution of Boussinesq.

In mathematics, Jacobi polynomialsP(α, β)
are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight (1 − x)α(1 + x)β on the interval [−1, 1]. The Gegenbauer polynomials, and thus also the Legendre, Zernike and Chebyshev polynomials, are special cases of the Jacobi polynomials.

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.


  1. 1 2 "The Definitive Glossary of Higher Mathematical Jargon — Indeterminate". Math Vault. 2019-08-01. Retrieved 2019-12-02.
  2. 1 2 Weisstein, Eric W. "Indeterminate". Retrieved 2019-12-02.
  3. 1 2 "Undefined vs Indeterminate in Mathematics". Retrieved 2019-12-02.
  4. Louis M. Rotando; Henry Korn (January 1977). "The indeterminate form 00". Mathematics Magazine. 50 (1): 41–42. doi:10.2307/2689754.
  5. "Table of equivalent infinitesimals" (PDF). Vaxa Software.