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indeterminate form

(Definition)

The expression

$\displaystyle \frac{0}{0}$

is known as the indeterminate form. The motivation for this name is that there are no rules for comparing the value of $\frac{0}{0}$ to the other real numbers. Note that, for example, $\frac{1}{0}$ is not indeterminate, since we can justifiably associate it with $+\infty$ , which does compare with the rest of the real numbers (in particular, it is defined to be greater than all of them.)

Other Indeterminate Forms

Although $\frac{0}{0}$ is often called “the” indeterminate form, there are many others. Some of these are:

$\frac{\infty}{\infty}$ , for the same motivating reasons as $\frac{0}{0}$ .
; which is the result of much impassioned debate (especially since is defined to be 1, counter-intuitively, but not unreasonably).
$1^{\infty}$ ; notably because of the derivation of :

$\displaystyle \lim_{n \to \infty} \left( 1+\frac{1}{n} \right)^n = e$

A direct substitution would yield $1^\infty$ .

"indeterminate form" is owned by akrowne.

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Other names:

indeterminate value

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Cross-references: derivation, associate, indeterminate, real numbers, expression
There are 8 references to this entry.

This is version 4 of indeterminate form, born on 2002-02-25, modified 2006-11-11.
Object id is 2658, canonical name is IndeterminateForm.
Accessed 5853 times total.

Classification:

AMS MSC:

12D99 (Field theory and polynomials :: Real and complex fields :: Miscellaneous)

Pending Errata and Addenda

None.[ View all 3 ]

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