Indeterminate form is a mathematical expression that can obtain any value depending on circumstances.
In calculus, it is usually possible to compute the 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.
and likewise for other arithmetic operations; this is sometimes called the algebraic limit theorem.
In these particular situations, the limit is said to take an indeterminate form, described by one of the informal expressions
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
A limit which unambiguously tends to infinity, for instance
[2] 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.
In each case, if the limits of the numerator and denominator are substituted, the resulting expression is
, 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.
is insufficient to determinate the limit An expression that arises by ways other than applying the algebraic limit theorem may have the same form of an indeterminate form.
Whether this expression is left undefined, or is defined to equal
, depends on the field of application and may vary between authors.
is particularly common in calculus, because it often arises in the evaluation of derivatives using their definition in terms of limit.
Other examples with this indeterminate form include and Direct substitution of the number that
approaches into any of these expressions shows that these are examples correspond to the indeterminate form
can also be obtained (in the sense of divergence to infinity): The following limits illustrate that the expression
is not commonly regarded as an indeterminate form, because if the limit of
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
) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.
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.
L'Hôpital's rule is a general method for evaluating the 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.
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
(the domain of logarithms is the set of all positive real numbers.)
, one of these forms may be more useful than the other in a particular case (because of the possibility of algebraic simplification afterwards).
The following table lists the most common indeterminate forms and the transformations for applying l'Hôpital's rule.