| Version current |
Version 3 |
| The expression |
The expression |
|
|
| $$ \frac{0}{0} $$ |
$$ \frac{0}{0} $$ |
|
|
| 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.) |
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 called ``the'' indeterminate form, another indeterminate form is |
|
|
| Although $\frac{0}{0}$ is often called ``the'' indeterminate form, there are many others. Some of these are: |
$$ \frac{\infty}{\infty} $$ |
|
|
| \begin{enumerate} |
for the same motivating reasons. |
|
|
| \item $ \frac{\infty}{\infty} $, for the same motivating reasons as $\frac{0}{0}$. |
Yet another is |
|
|
| \item $ 0^0 $; which is the result of much impassioned debate (especially since $0!$ is defined to be 1, counter-intuitively, but not unreasonably). |
$$ 0^0 $$ |
|
|
| \item $1^{\infty}$; notably because of the derivation of $e$: |
which is the result of much impassioned debate (especially since $0!$ is defined to be 1, counter-intuitively, but not unreasonably). |
|
|
| $$ \lim_{n \to \infty} \left( 1+\frac{1}{n} \right)^n = e $$ |
|
|
|
| A direct substitution would yield $1^\infty$. |
|
|
|
| \end{enumerate} |
|
