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 N2–3Definition
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.