CONTINUITY
Fig. 7-5
7.10
If the function
is continuous, what is the value of C?
Since x
2 - 16 = (x - 4)(x + 4), /(*) = x + 4 for x ¥= 4. Hence, lim f(x) = 8. For continuity, we
must have lim /(*) =/(4), and, therefore, 8 = /(4)=C.
7.11
Let g(x) be the function such that
(«) = ° by definition, (fc) Since x2 - b2 = (* - b)(x + b), g(x) = for x^b. Hence,
lim g(x)= lim(x + b) = 2b. (c) For g(x) to be continuous at x = b, we must have lim g(x) = g(b), that is,
x — >b
x — *b
x — *b
2b = 0. So, g(x) is continuous at x = b only when b = 0.
(a) Does g(b) exist? (b) Does lim g(x) exist? (c) Is g(x) continuous at x = fc?
JC-*fe
Notice that
and
are defined when
since
and
are nonnegative for
Also,
Hence,
Therefore, k must be §.
7.12
For what value of k is the following a continuous function?
and
45
if
if
if
if
7x+2
6x+4
x=2