Indeterminate form: Difference between revisions

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{{Short description|Expression in mathematical analysis}}

In [[calculus]] and other branches of [[mathematical analysis]], when you take the [[limit (mathematics)|limit]] of the sum, difference, product, quotient or power of two functions, you may often be able to simply add, subtract, multiply, divide or exponentiate the corresponding limits of these two functions respectively. However, there are occasions where it is unclear what the sum, difference, product or power of these two limits ought to be. For example, it is unclear what the following expressions ought to evaluate to:<ref name=":1" />

In [[calculus]] and other branches of [[mathematical analysis]], limits involving an arithmetic combination of functions in an independent variable may often be evaluated by replacing these functions by their [[limit (mathematics)|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 <math>0~</math>, <math>1</math> or <math>\infty</math>, 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).<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Indeterminate.html|title=Indeterminate|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}</ref> 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:<ref name=":1" />

:<math>\frac 00,~ \frac{\infty}{\infty},~ 0\times\infty,~ \infty - \infty,~ 0^0,~ 1^\infty, \text{ and } \infty^0 .</math>

InThese [[calculus]]seven andexpressions otherare branchesknown ofas [[mathematical'''indeterminate analysis]],forms'''. limitsMore involvingspecifically, ansuch arithmeticexpressions combinationare of functions in an independent variable may often be evaluatedobtained by replacingnaively theseapplying functions by theirthe [[algebraic limit (mathematics)|limitstheorem]]; if the expression obtained after this substitution does not provide sufficient information to determineevaluate the original limit, thenof the expressioncorresponding isarithmetic calledoperation anof '''indeterminatetwo form'''.functions, Moreyet specifically,there anare indeterminateexamples formof ispairs aof mathematicalfunctions expressionthat involvingafter atbeing mostoperated twoon ofconverge to <math>0~</math>, <math>1</math>converge orto <math>\infty</math>,another obtainedfinite byvalue, applyingdiverge theto [[algebraicinfinity limitor theorem]]just indiverge. theThis processinability ofto attemptingdecide towhat determine athe limit, which failsought to restrictbe thatexplains limitwhy tothese oneforms specificare valueregarded oras infinity, and thus does not determine the limit being sought'''indeterminate'''. A limit confirmed to be infinity is not indeterminate since it has been determined to have a specific value (infinity).<ref name=":1">{{Cite web|url=http://mathworld.wolfram.com/Indeterminate.html|title=Indeterminate|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-02}}</ref> 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 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 <math>0/0</math>". For example, as <math>x</math> approaches <math>0~</math>, the ratios <math>x/x^3</math>, <math>x/x</math>, and <math>x^2/x</math> go to <math>\infty</math>, <math>1</math>, and <math>0~</math> respectively. In each case, if the limits of the numerator and denominator are substituted, the resulting expression is <math>0/0</math>, which is undefined. In a loose manner of speaking, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, and it is easy to construct similar examples for which the limit is any particular value.