# indeterminate form

 $\frac{0}{0}$

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. 1.

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

2. 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. 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 $1^{\infty}$.

Title indeterminate form IndeterminateForm 2013-03-22 12:28:19 2013-03-22 12:28:19 akrowne (2) akrowne (2) 7 akrowne (2) Definition msc 12D99 indeterminate value LHpitalsRule ImproperLimits EmptyProduct