indeterminate formIndeterminateForm2002-02-25 01:08:152006-11-11 10:17:52Definition\usepackage{amssymb}
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%\usepackage{xypic}The expression
$$ \frac{0}{0} $$
is known as the \emph{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 \emph{not} indeterminate, since we can justifiably associate it with $+\infty$, which \emph{does} compare with the rest of the real numbers (in particular, it is defined to be greater than all of them.)
\section{Other Indeterminate Forms}
Although $\frac{0}{0}$ is often called ``the'' indeterminate form, there are many others. Some of these are:
\begin{enumerate}
\item $ \frac{\infty}{\infty} $, for the same motivating reasons as $\frac{0}{0}$.
\item $ 0^0 $; which is the result of much impassioned debate (especially since $0!$ is defined to be 1, counter-intuitively, but not unreasonably).
\item $1^{\infty}$; notably because of the derivation of $e$:
$$ \lim_{n \to \infty} \left( 1+\frac{1}{n} \right)^n = e $$
A direct substitution would yield $1^\infty$.
\end{enumerate}