# Free Variable/Examples

## Examples of Free Variables

### Calculus Example

$\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$

$x$ is a free variable, as a function's derivative varies with the input being considered.

### Cardinality Example

In set theory:

$\card S = \aleph_0$

$S$ is a free variable, as, for instance, $S = \Z$ makes this true while $S = \R$ makes it false.

### Series Example

In the inequality:

$\ds \sum_{n \mathop = 0}^\infty a z^n < z^2$

$a$ and $z$ are both free variables, as the inequality may or may not hold depending on their values.