Indeterminate form: Difference between revisions

Revision as of 09:22, 22 July 2023

Revision as of 05:29, 5 August 2023

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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 is taken, youit may often be ablepossible 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" />

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

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These seven expressions are known as '''indeterminate forms'''. More specifically, such expressions are obtained by naively applying the [[algebraic limit theorem]] to evaluate the limit of the corresponding arithmetic operation of two functions, yet there are examples of pairs of functions that after being operated on converge to 0, converge to another finite value, diverge to infinity or just diverge. This inability to decide what the limit ought to be explains why these forms are regarded as '''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 is the quotient of two functions each of which converges to zero. This indeterminate form is denoted by <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 indeterminate. In this sense, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, by appropriate choices of functions to put in the numerator and denominator. You may in fact find aA pair of functions for which the limit is any particular given value may in fact be found. Even more surprising, perhaps, the quotient of the two functions may in fact diverge, and not merely diverge to infinity. For example, <math> x \sin(1/x) / x</math>.

So the fact that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> converge to <math>0~</math> as <math>x</math> approaches some [[limit point]] <math>c</math> is insufficient to determinate the [[limit of a function|limit]]