Wikipedia / Indeterminate form
Wikipedia / Indeterminate form
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. For example,
and likewise for other arithmetic operations; this is sometimes called the 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
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 or as indicated.
is not considered indeterminate. The term was originally introduced by Cauchy's student Moigno in the middle of the 19th century.
.
So the fact that two functions approaches some limit point is insufficient to determine the limit
, depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero and other expressions involving infinity are not indeterminate forms.
Some examples and non-examples
Indeterminate form 0/0
is particularly common in calculus, because it often arises in the evaluation of derivatives using their definition in terms of limit.
As mentioned above, while is an indeterminate form. Other examples with this indeterminate form include and Direct substitution of the number that can be obtained for this indeterminate form as follows: can also be obtained (in the sense of divergence to infinity):
Indeterminate form 00
are analytic. Otherwise, use the transformation in the table below to evaluate the limit.
Expressions that are not indeterminate forms
exists so that:
The limit fails to exist.
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).) is not an indeterminate form, since a quotient giving rise to such an expression will always diverge.
.
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.
Equivalent infinitesimal
, they are called equivalent infinitesimal).
Here is a brief proof:
.
, one can make use of the following facts about equivalent infinitesimals if x becomes closer to zero):
For example:
as y is used in the 5th equality.
L'Hôpital's rule
. This rule states that (under appropriate conditions)
are the derivatives. (Note that this rule does not, 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:
, 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 is asymptotically positive. (the domain of logarithms is the set of all positive real numbers.)
.
List of indeterminate forms
The following table lists the most common indeterminate forms and the transformations for applying l'Hôpital's rule.