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, counterintuitively, 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 ${1}^{\mathrm{\infty}}$.
Title  indeterminate form 

Canonical name  IndeterminateForm 
Date of creation  20130322 12:28:19 
Last modified on  20130322 12:28:19 
Owner  akrowne (2) 
Last modified by  akrowne (2) 
Numerical id  7 
Author  akrowne (2) 
Entry type  Definition 
Classification  msc 12D99 
Synonym  indeterminate value 
Related topic  LHpitalsRule 
Related topic  ImproperLimits 
Related topic  EmptyProduct 