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.)

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 $1^\infty$ .