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# indeterminate form

The expression

$\frac{0}{0}$ |

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.)

# 1 Other Indeterminate Forms

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

1. $\frac{\infty}{\infty}$, for the same motivating reasons as $\frac{0}{0}$.

2. $0^{0}$; which is the result of much impassioned debate (especially since $0!$ is defined to be 1, counter-intuitively, but not unreasonably).

3. $1^{{\infty}}$; notably because of the derivation of $e$:

$\lim_{{n\to\infty}}\left(1+\frac{1}{n}\right)^{n}=e$ A direct substitution would yield $1^{\infty}$.

## Mathematics Subject Classification

12D99*no label found*

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## Comments

## "See also"

Hi, if I add to the "See also" of one of my entries (now "improper limits") e.g. the name of akrowne's entry "IndeterminateForm", the name of my entry does not reciprocally appear in the "See also" of this latter entry (as it would have appeared earlier). Has the system been altered?

Jussi

## Re: "See also"

As far as I know the system hasn't been altered. Reciprocal links are not always created - it depends on the other owner's preference settings.