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 $+\mathrm{\infty }$, 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. $\frac{\mathrm{\infty }}{\mathrm{\infty }}$, 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. $1^{\mathrm{\infty }}$; notably because of the derivation of $e$:

   $$\underset{n\to \mathrm{\infty }}{lim}{\left(1+\frac{1}{n}\right)}^n=e$$

   A direct substitution would yield $1^{\mathrm{\infty }}$.