
Limits and Continuity 3-41
=
+ +
+
= =
≠
→
( )
( )
lim ( ) ( ).
x x
x
f x f
x
2
3
3 9
3
27
6
9
2
3i.e.,
Therefore, f is not continuous at x = 3.
Now, let us check for continuity at x = -3.
Function f is not defined at x = -3 because of division by zero. Thus, f(-3) does not exist, and hence
f is NOT continuous at x = -3.
Example 3.39 Determine the values of x at which f is continuous if f is given by
f x
e
x
x
( ) .
cos
=
− −3 16
2
Solution: First, we describe the numerator of the function f using functional composition.
Let g(x) = cos x and h(x) = e
x
.
Clearly, both the functions cos x and e
x
are continuous for all values of x. Since the composition of
continuous functions ...