Answers and Explanations 223
264.
continuous at a = 2, jump discontinuity at a = 3
A function f
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
To determine whether the function is continuous at a = 2, see whether it satisfies the
equation lim x f x
f
→
( ) = ( )
2
2 . The left-hand limit at a = 2 is lim
x
x
→
−
− =
− =
2
2
2 2 0, and the
right-hand limit at a = 2 is lim
x
x
→
+
−
(
) = ( ) − =
2
2
2
4
2
4 0. Because f 2
2 2 0
( ) = − = , the function is continuous at a = 2.
Likewise, decide whether the function satisfies the equation lim x f x
f
→
( ) = ( )
3
3 .
The left-hand limit at a = 3 is lim
x
x
→
−
−
(
) = ( ) − =
3
2
2
4
3
4 5, and the right-hand limit
at a = 3 is lim
x
x
→
+
+
( ) = +
=
3
1
5
1
3 5
1
8
. The limits don’t match, so there’s a jump
discontinuity at a = 3.
265.
infinite discontinuity at a = 0, continuous at a = 4
A function f
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
To determine whether the function is continuous at a = 0, see whether it satisfies
the equation lim x f x
f
→
( ) = ( )
0
0 . The left-hand limit at a = 0 is lim cos
cos
x
x
→
−
=
=
0
0 1, and
the right-hand limit at a = 0 is lim
x
x
→
+
+
→
→∞
0
1
1
0
, so there’s an infinite discontinuity at a = 0.
Likewise, decide whether the function satisfies the equation lim x f
f
→
( ) = ( )
4
4
4 . The
left-hand limit at a = 4 is lim
x
x
→
−
=
4
1 1
4
, and the right-hand limit at a = 4 is
lim
x
x
→
+
+
= +
= =
4
2
4
2
4 4
2
8
1
4
. Because f 4
1
4
( ) = , the function is continuous at a = 4.
266.
c = − 3
2
To determine the value of c, you must satisfy the definition of a continuous function:
lim x
f x
f
→
( ) = ( )
2
2 .
The left-hand limit at x = 2 is given by lim
x
cx
c
c
→
−
−
(
)= ( )− = −
2
2
2 2 2 2, and the right-hand
limit is given by lim
x
cx
c
c
→
+
+
(
) = ( ) + = +
2
2
2
1
2
1 4 1. Also note that f
(2) = 2c – 2.
The left-hand limit must equal the right-hand limit, so set them equal to each other and
solve for c:
2 2 4 1
3 2
3
2
c
c
c
c
− = +
− =
− =
Therefore, c = − 3
2
is the solution.
267.
c = 2
To determine the value of c, you must satisfy the definition of a continuous function:
lim x
f x
f
→
( ) = ( )
4
4 .
Answers
201–300
264.
continuous at a = 2, jump discontinuity at a = 3
A function f
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
To determine whether the function is continuous at a = 2, see whether it satisfies the
equation lim x f x
f
→
( ) = ( )
2
2 . The left-hand limit at a = 2 is lim
x
x
→
−
− =
− =
2
2
2 2 0, and the
right-hand limit at a = 2 is lim
x
x
→
+
−
(
) = ( ) − =
2
2
2
4
2
4 0. Because f 2
2 2 0
( ) = − = , the function is continuous at a = 2.
Likewise, decide whether the function satisfies the equation lim x f x
f
→
( ) = ( )
3
3 .
The left-hand limit at a = 3 is lim
x
x
→
−
−
(
) = ( ) − =
3
2
2
4
3
4 5, and the right-hand limit
at a = 3 is lim
x
x
→
+
+
( ) = +
=
3
1
5
1
3 5
1
8
. The limits don’t match, so there’s a jump
discontinuity at a = 3.
265.
infinite discontinuity at a = 0, continuous at a = 4
A function f
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
To determine whether the function is continuous at a = 0, see whether it satisfies
the equation lim x f x
f
→
( ) = ( )
0
0 . The left-hand limit at a = 0 is lim cos
cos
x
x
→
−
=
=
0
0 1, and
the right-hand limit at a = 0 is lim
x
x
→
+
+
→
→∞
0
1
1
0
, so there’s an infinite discontinuity at a = 0.
Likewise, decide whether the function satisfies the equation lim x f
f
→
( ) = ( )
4
4
4 . The
left-hand limit at a = 4 is lim
x
x
→
−
=
4
1 1
4
, and the right-hand limit at a = 4 is
lim
x
x
→
+
+
= +
= =
4
2
4
2
4 4
2
8
1
4
. Because f 4
1
4
( ) = , the function is continuous at a = 4.
266.
c = − 3
2
To determine the value of c, you must satisfy the definition of a continuous function:
lim x
f x
f
→
( ) = ( )
2
2 .
The left-hand limit at x = 2 is given by lim
x
cx
c
c
→
−
−
(
)= ( )− = −
2
2
2 2 2 2, and the right-hand
limit is given by lim
x
cx
c
c
→
+
+
(
) = ( ) + = +
2
2
2
1
2
1 4 1. Also note that f
(2) = 2c – 2.
The left-hand limit must equal the right-hand limit, so set them equal to each other and
solve for c:
2 2 4 1
3 2
3
2
c
c
c
c
− = +
− =
− =
Therefore, c = − 3
2
is the solution.
267.
c = 2
To determine the value of c, you must satisfy the definition of a continuous function:
lim x
f x
f
→
( ) = ( )
4
4 .
Answers
201–300