Prove that .

SOLUTION: The graph of |x| is shown in Figure N2–3.

We examine both left- and right-hand limits of the absolute-value function as x

→ 0. Since

it follows that

Since the left-hand and right-hand limits both equal 0, .

Note that but equals −c if c < 0.

`Figure N2–3`

###### Definition

The function f(x) is said to become infinite (positively or negatively) as x

approaches c if f (x) can be made arbitrarily large (positively or negatively) by

taking x sufficiently close to c. We write

Since for the limit to exist it must be a finite number, neither of the preceding

limits exists.

This definition can be extended to include x approaching c from the left or

from the right. The following examples illustrate these definitions.