Answers and Explanations 219
253.
removable discontinuity at x = 1, infinite discontinuity at x = 5
The limit exists at x = 1, but f
(1) is undefined, which corresponds to a removable
discontinuity.
At x = 5, the left-hand limit is ∞ and the right-hand limit is –∞, so an infinite discontinuity exists at x = 5.
254.
jump discontinuity at x = –2, jump discontinuity at x = 3
At x = –2, the left-hand limit doesn’t equal the right-hand limit (both limits exist as
finite values), which corresponds to a jump discontinuity.
At x = 3, the left-hand limit again doesn’t equal the right-hand limit (both limits exist as
finite values), so this also corresponds to a jump discontinuity.
255.
removable discontinuity at x = –1, jump discontinuity at x = 4, infinite discontinuity at x = 6
At x = –1, the left-hand limit equals the right-hand limit, but the limit doesn’t equal f
(–1),
which is undefined. Therefore, a removable discontinuity is at x = –1.
At x = 4, the left-hand limit doesn’t equal the right-hand limit (both limits exist as finite
values), so a jump discontinuity is at x = 4.
At x = 6, both the left and right-hand limits equal ∞, so an infinite discontinuity is at x = 6.
256.
not continuous, infinite discontinuity
A function f
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
The left-hand limit at a is given by lim
x
x
→
−
−
2
1
2
. As x → 2
–
, you have (x – 2) → 0
–
so
that 1
2
1
0
x −
→
→−∞
−
. Because the discontinuity is infinite, you don’t need to
examine the right-hand limit; you can conclude that the function is not continuous.
257.
continuous, f
(a) = 2
A function f
(x) is continuous at x = a if it satisfies the equation lim x a
f x
f a
→
( ) = ( ).
The left-hand limit at a is lim
x
x
→
−
+
(
) = + =
1
2
2
1
1 1 2, and the right-hand limit at a is
lim
x
x
→
+
−
(
) = − =
1
4
2 4 1 2 2. The left-hand and right-hand limits match, so the limit
at a exists and is equal to 2.
The value at a is f a
f
( ) = ( ) =
− =
1 4 1 2 2. Because the function satisfies the definition
of continuity, you can conclude that that function is continuous at a = 1.
258.
continuous, f
(a) = 5
A function f
(x) is continuous at x = a if it satisfies the equation lim x a
f x
f a
→
( ) = ( ).
The left-hand limit at a is
Answers
201–300
253.
removable discontinuity at x = 1, infinite discontinuity at x = 5
The limit exists at x = 1, but f
(1) is undefined, which corresponds to a removable
discontinuity.
At x = 5, the left-hand limit is ∞ and the right-hand limit is –∞, so an infinite discontinuity exists at x = 5.
254.
jump discontinuity at x = –2, jump discontinuity at x = 3
At x = –2, the left-hand limit doesn’t equal the right-hand limit (both limits exist as
finite values), which corresponds to a jump discontinuity.
At x = 3, the left-hand limit again doesn’t equal the right-hand limit (both limits exist as
finite values), so this also corresponds to a jump discontinuity.
255.
removable discontinuity at x = –1, jump discontinuity at x = 4, infinite discontinuity at x = 6
At x = –1, the left-hand limit equals the right-hand limit, but the limit doesn’t equal f
(–1),
which is undefined. Therefore, a removable discontinuity is at x = –1.
At x = 4, the left-hand limit doesn’t equal the right-hand limit (both limits exist as finite
values), so a jump discontinuity is at x = 4.
At x = 6, both the left and right-hand limits equal ∞, so an infinite discontinuity is at x = 6.
256.
not continuous, infinite discontinuity
A function f
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
The left-hand limit at a is given by lim
x
x
→
−
−
2
1
2
. As x → 2
–
, you have (x – 2) → 0
–
so
that 1
2
1
0
x −
→
→−∞
−
. Because the discontinuity is infinite, you don’t need to
examine the right-hand limit; you can conclude that the function is not continuous.
257.
continuous, f
(a) = 2
A function f
(x) is continuous at x = a if it satisfies the equation lim x a
f x
f a
→
( ) = ( ).
The left-hand limit at a is lim
x
x
→
−
+
(
) = + =
1
2
2
1
1 1 2, and the right-hand limit at a is
lim
x
x
→
+
−
(
) = − =
1
4
2 4 1 2 2. The left-hand and right-hand limits match, so the limit
at a exists and is equal to 2.
The value at a is f a
f
( ) = ( ) =
− =
1 4 1 2 2. Because the function satisfies the definition
of continuity, you can conclude that that function is continuous at a = 1.
258.
continuous, f
(a) = 5
A function f
(x) is continuous at x = a if it satisfies the equation lim x a
f x
f a
→
( ) = ( ).
The left-hand limit at a is
Answers
201–300