Sullivan Microsite DigiSample

Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.2 • Assess Your Understanding 101
In Problems 73–80, find lim f (x) and lim f (x) for the
x→c− x→c +
given number c. Based on the results, determine whether lim f (x)
x→c
exists.
2x − 3 if x ≤ 1
PAGE
93 73. f (x) =
3 − x if x > 1
at c = 1
5x + 2 if x < 2
74. f (x) =
1 + 3x if x ≥ 2
at c = 2
⎧
⎨ 3x − 1 if x < 1
75. f (x) = 4 if x = 1 at c = 1
⎩
2x if x > 1
⎧
⎨ 3x − 1 if x < 1
76. f (x) = 2 if x = 1 at c = 1
⎩
2x if x > 1
x − 1 if x < 1
77. f (x) = √
x − 1 if x > 1
at c = 1
78. f (x) =
9 − x
2 if 0 < x < 3
at c = 3
x 2 − 9 if x > 3
79.
⎧
⎨ x 2 − 9
if x = 3
f (x) = x − 3
⎩
6 if x = 3
at c = 3
⎧
⎨ x − 2
if x = 2
80. f (x) = x 2 − 4
at c = 2
⎩
1 if x = 2
Applications and Extensions
Heaviside Functions In Problems 81 and 82, find the limit, if it
exists, of the given Heaviside function at c.
0 if t < 1
PAGE
94 81. u 1(t) =
1 if t ≥ 1
at c = 1
0 if t < 3
82. u 3(t) =
1 if t ≥ 3
at c = 3
In Problems 83–92, find each limit.
(x + h) 2 − x 2
83. lim h→0 h
1
85. lim
x + h − 1 x
h→0 h
1 1
87. lim x→0 x 4 + x − 1
4
√ √ x + h − x
84. lim h→0 h
1
(x + h)
86. lim
3 − 1 x 3
h→0
h
2
88. lim
x→−1 x + 1
x − 7
x − 2
89. lim √ 90. lim √ x→7 x + 2 − 3 x→2 x + 2 − 2
91. lim x→1
x 3 − 3x 2 + 3x − 1
x 2 − 2x + 1
1
3 − 1
x + 4
x 3 + 7x 2 + 15x + 9
92. lim
x→−3 x 2 + 6x + 9
93. Cost of Water The Jericho Water District determines quarterly
water costs, in dollars, using the following rate schedule:
Water used
(in thousands of gallons) Cost
0 ≤ x ≤ 10 $9.00
10 < x ≤ 30 $9.00 + 0.95 for each thousand
gallons in excess of 10,000 gallons
30 < x ≤ 100 $28.00 + 1.65 for each thousand
gallons in excess of 30,000 gallons
x > 100 $143.50 + 2.20 for each thousand
gallons in excess of 100,000 gallons
Source: Jericho Water District, Syosset, NY.
(a) Find a function C that models the quarterly cost, in dollars, of
using x thousand gallons of water.
(b) What is the domain of the function C?
(c) Find each of the following limits. If the limit does not exist,
explain why.
lim C(x) lim C(x)
x→5 x→10
(d) What is lim C(x)?
x→0 +
(e) Graph the function C.
lim C(x)
x→30
lim C(x)
x→100
94. Cost of Electricity In June 2016, Florida Power and Light had
the following monthly rate schedule for electric usage in
single-family residences:
Monthly customer charge $7.87
Fuel charge
≤ 1000 kWH $0.02173 per kWH
> 1000 kWH $21.73 + 0.03173 for each kWH
in excess of 1000
Source: Florida Power and Light, Miami, FL.
(a) Find a function C that models the monthly cost, in dollars, of
using x kWH of electricity.
(b) What is the domain of the function C?
(c) Find lim C(x), if it exists. If the limit does not exist,
x→1000
explain why.
(d) What is lim C(x)?
x→0 +
(e) Graph the function C.
95. Low-Temperature Physics In thermodynamics, the average
molecular kinetic energy (energy of motion) of a gas having
molecules of mass m is directly proportional to its temperature T
on the absolute (or Kelvin) scale. This can be expressed as
1
2 mv2 = 3 kT, where v = v(T ) is the speed of a typical molecule
2
at time t and k is a constant, known as the Boltzmann constant.
(a) What limit does the molecular speed v approach as the gas
temperature T approaches absolute zero (0 K or −273 ◦ C
or −469 ◦ F)?
(b) What does this limit suggest about the behavior of a gas as
its temperature approaches absolute zero?
88. 2 9
89. 6
90. 4
91. 0
92. –2
93. (a)
⎧
⎪
⎪
Cx ( ) = ⎨
⎪
⎪
⎩
9.00 0≤x
≤10
9.00 + 0.95( x− 10) 10 < x ≤30
28.00 + 1.65( x− 30) 30 < x ≤100
143.50 + 2.20( x− 100) x > 100
(b) { xx≥
0}
(c)
lim Cx ( ) = 9.00, lim Cx ( ) = 9.00,
x→5
lim Cx ( ) = 28.00,
x→30
(d) 9.00
(e)
94. (a)
Quarterly water cost
(dollars)
y
250
200
150
100
50
x→10
lim Cx ( ) = 143.50
x→100
50 100 150 x
Water usage (thousands of gallons)
⎪
x
( ) = ⎨
( x )
C x
⎧ 7.87 + 0.02173 if 0 ≤x
≤ 1000
+ − >
⎩
⎪
29.60 0.3173 1000 if x 1000
(b) { xx≥
0}
(c) lim Cx ( ) = 8.9565;
−
x→50
lim Cx ( ) = 8.9565;
+
x→
50
lim Cx ( ) = 8.9565
x→50
(d) lim Cx ( ) = 7.87
x→ 0
+
73. lim fx ( ) =−1,
lim fx ( ) = 2,
x→1
−
not exist.
74. lim fx ( ) = 12,
x→2
−
not exist.
75. lim fx ( ) = 2,
x→1
−
76. lim fx ( ) = 2,
x→1
−
77. lim fx ( ) = 0,
x→1
−
78. lim fx ( ) = 0,
x→3
−
x→ 1
+
lim fx ( ) = 7,
x→ 2
+
lim fx ( ) does
x→1
lim fx ( ) does
x→ 2
lim fx ( ) = 2, lim fx ( ) = 2
x→ 1
+
x→1
lim fx ( ) = 2, lim fx ( ) = 2
x→ 1
+
x→1
lim fx ( ) = 0, lim fx ( ) = 0
x→ 1
+
x→1
lim fx ( ) = 0, lim fx ( ) = 0
x→ 3
+
x→3
79. lim fx ( ) = 6,
x→3
−
lim fx ( ) = 6,
x→ 3
+
1
80. lim fx ( ) =
4 , lim fx ( )
x→2
−
x→ 2
+
81. Limit does not exist.
82. Limit does not exist.
83. 2x
1
84.
2 x
1
85. −
x
2
86. −
x
3
4
87. − 1
16
=
1
4 ,
lim fx ( ) = 6
x→3
lim fx ( ) =
x→2
1
4
(e)
C
50
40
30
20
10
95. (a) 0
1 2 3 4 5 6 7 8
(b) As the temperature of a gas
approaches zero, the molecules in
the gas stop moving.
x
Section 1.2 • Assess Your Understanding
101
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