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What is indeterminate form

  1. Jul 23, 2014 #1
    Definition/Summary

    An algebraic function of a pair of numbers is an indeterminate form at a particular pair of values (which may include infinity) if the function does not tend to a unique limit at that pair of values.

    For example, [itex]\infty\ -\ \infty[/itex] is an indeterminate form because the function [itex]f(x - y)[/itex] does not tend to a unique limit at the values [itex]x\ =\ y\ =\ \infty[/itex].

    Equations

    Commonly encountered indeterminate forms:
    [tex]0/0,~ \infty/\infty,~0\cdot \infty,~0^0,~1^{\infty},~\infty ^0,~\infty-\infty[/tex]

    Commonly encountered forms which are not indeterminate:
    [tex]1/0,~0/\infty,~\infty/0,~0^{\infty}[/tex]

    Extended explanation

    Examples:

    1. Consider the form [itex]\infty-\infty[/itex], which can be made from the limit as [itex]x\rightarrow \infty[/itex] of [itex]x^2-x[/itex], [itex]x-x[/itex] or [itex]x-x^2[/itex]. In these three cases, we find the limits are [itex]\infty[/itex], [itex]0[/itex] and [itex]-\infty[/itex] respectively.

    2. Consider the form [itex]\infty /\infty[/itex], which can be made from the limit as [itex]x\rightarrow \infty[/itex] of [itex]x^2/x[/itex], [itex]x/x[/itex] or [itex]x/x^2[/itex]. In these three cases, we find the limits are [itex]\infty[/itex], [itex]1[/itex] and [itex]0[/itex] respectively.

    The lack of uniqueness makes these forms indeterminate.

    l'Hôpital's rule:

    If [itex]f(x)\ =\ g(x)/h(x)[/itex] and [itex]g(a)/h(a)[/itex] is an indeterminate form, [itex]0/0[/itex] or [itex]\infty/\infty[/itex], at some value [itex]a[/itex], but [itex]g'(a)/h'(a)[/itex] is not, then l'Hôpital's rule can be used to find the limiting value [itex]f(a)[/itex]:

    [tex]\lim_{x \rightarrow a} \frac{g(x)}{h(x)}
    = \lim_{x \rightarrow a} \frac{g'(x)}{h'(x)}[/tex]

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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