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 at most two of {\displaystyle 0~}{\displaystyle 0~}, {\displaystyle 1}1 or {\displaystyle \infty }\infty , 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).[1] 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:[1]
{\displaystyle {\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,1^{\infty },{\text{ and }}\infty ^{0}.}{\displaystyle {\frac {0}{0}},~{\frac {\infty }{\infty }},~0\times \infty ,~\infty -\infty ,1^{\infty },{\text{ and }}\infty ^{0}.}
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 {\displaystyle 0/0}0/0”. For example, as {\displaystyle x}x approaches {\displaystyle 0~}{\displaystyle 0~}, the ratios {\displaystyle x/x^{3}}{\displaystyle x/x^{3}}, {\displaystyle x/x}{\displaystyle x/x}, and {\displaystyle x^{2}/x}{\displaystyle x^{2}/x} go to {\displaystyle \infty }\infty , {\displaystyle 1}1, and {\displaystyle 0~}{\displaystyle 0~} respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is {\displaystyle 0/0}0/0, which is undefined. In a loose manner of speaking, {\displaystyle 0/0}0/0 can take on the values {\displaystyle 0~}{\displaystyle 0~}, {\displaystyle 1}1, or {\displaystyle \infty }\infty , and it is easy to construct similar examples for which the limit is any particular value.
So, given that two functions {\displaystyle f(x)}f(x) and {\displaystyle g(x)}g(x) both approaching {\displaystyle 0~}{\displaystyle 0~} as {\displaystyle x}x approaches some limit point {\displaystyle c}c, that fact alone does not give enough information for evaluating the limit
{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}.}{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}.}
Not every undefined algebraic expression corresponds to an indeterminate form.[2] For example, the expression {\displaystyle 1/0}1/0 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, {\displaystyle 0/0}0/0 which arises from substituting {\displaystyle 0~}{\displaystyle 0~} for {\displaystyle x}x in the equation {\displaystyle f(x)=|x|/(|x-1|-1)}{\displaystyle f(x)=|x|/(|x-1|-1)} 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 {\displaystyle 0{0}. Whether this expression is left undefined, or is defined to equal {\displaystyle 1}1, depends on the field of application and may vary between authors. For more, see the article Zero to the power of zero. Note that {\displaystyle 0^{\infty }}{\displaystyle 0^{\infty }} and other expressions involving infinity are not indeterminate forms.
Contents
1 Some examples and non-examples
1.1 Indeterminate form 0/0
1.2 Indeterminate form 00
1.3 Expressions that are not indeterminate forms
2 Evaluating indeterminate forms
2.1 Equivalent infinitesimal
2.2 L’Hôpital’s rule
3 List of indeterminate forms
4 See also