FIGURE N2–2
SOLUTIONS:
(a) so the right-hand limit exists at x = −2, even though f is not defined at x = −2.
(b) does not exist. Although f is defined at x = 0 (f (0) = 2), we observe that
whereas For the limit to exist at a point, the left-hand and right-hand limits must be
the same.
(c) This limit exists because Indeed, the limit exists at x = 2
even though it is different from the value of f at 2 (f (2) = 0).
(d) so the left-hand limit exists at x = 4.
EXAMPLE 3
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 if c > 0 but equals −c if c < 0.