In
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
and other branches of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, 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 at most two of
,
or
, 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, and thus does not determine the limit being sought. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity).
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:
:
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
Limit or Limits may refer to:
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Not every undefined algebraic expression corresponds to an indeterminate form.
For example, the expression
is undefined as a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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
File:Indeterminate form - x over x.gif, Fig. 1: =
File:Indeterminate form - x2 over x.gif, Fig. 2: =
File:Indeterminate form - sin x over x close.gif, Fig. 3: =
File:Indeterminate form - complicated.gif, Fig. 4: = (for = 49)
File:Indeterminate form - 2x over x.gif, Fig. 5: = where = 2
File:Indeterminate form - x over x3.gif, Fig. 6: =
The indeterminate form
is particularly common in
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, because it often arises in the evaluation of
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s using their definition in terms of limit.
As mentioned above,
while
This is enough to show that
is an indeterminate form. 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
, but these limits can assume many different values. Any desired value
can be obtained for this indeterminate form as follows:
The value
can also be obtained (in the sense of divergence to infinity):
Indeterminate form 00
File:Indeterminate form - x0.gif, Fig. 7: =
File:Indeterminate form - 0x.gif, Fig. 8: = 0
The following limits illustrate that the expression
is an indeterminate form:
Thus, in general, knowing that
and
is not sufficient to evaluate the limit
If the functions
and
are
analytic
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* ...
at
, and
is positive for
sufficiently close (but not equal) to
, then the limit of
will be
. 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:
#
approaches
#
approaches
# 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
). 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.
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
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
s (e.g.,
if ''x'' becomes closer to zero):
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
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s 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 0
0:
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
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
(ln) is a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
; 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 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.
See also
*
Defined and undefined
*
Division by zero
*
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
*
Indeterminate equation
*
Indeterminate system
*
Indeterminate (variable)
In mathematics, particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else except itself. It may be used as a placeholder in objects such as polynomials and formal power seri ...
*
L'Hôpital's rule
References
{{Calculus topics
Limits (mathematics)