Barrons AP Calculus

(a)

(b)

(c)

(d)

Determine whether limits of f, if any, exist at

(a) x   =   −2, (b) x   =   0,

(c) x   =   2, (d)  x   =   4.

Figure  N2–2

SOLUTIONS:

,   so  the right-hand  limit   exists  at  x   =   −2, even    though  f   is  not

defined at  x   =   −2.

    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.

. 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).
, so the left-hand limit exists at x = 4.

Example 3 __