Indeterminate form: Difference between revisions

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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 occurs when determiningis the limit of the ratioquotient of two functions, in which botheach of thesewhich functions tendconverges to zero. inThis theindeterminate limit, andform is referreddenoted to as "the indeterminate formby <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 undefinedindeterminate. In a loose manner ofthis speakingsense, <math>0/0</math> can take on the values <math>0~</math>, <math>1</math>, or <math>\infty</math>, andby itappropriate ischoices easyof functions to constructput similarin examplesthe numerator and denominator. You may in fact find a pair of functions for which the limit is any particular given value. 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, given that two [[function (mathematics)|functions]] <math>f(x)</math> and <math>g(x)</math> both approaching <math>0~</math> as <math>x</math> approaches some [[limit point]] <math>c</math>, that fact alone does not give enough information for evaluating the [[limit of a function|limit]]