1-3 Continuity, End Behavior, and Limits
Use logical reasoning to determine the end behavior or limit of the function as x approaches infinity.
Explain your reasoning.
33. SOLUTION: Sample answer: As x
, the fraction will decrease, and q(x) will approach 0.
35. p (x) =
SOLUTION: Sample answer: As x
, the fraction will approach
, so p (x) will approach 1.
37. c(x) =
SOLUTION: Sample answer: As x
, the fraction will decrease, and c(x) will approach 0.
39. h(x) = 2x5 + 7x3 + 5
SOLUTION: Sample answer: As x
, the h(x) will increase without bound, or approach ∞.
41. PHYSICS The kinetic energy of an object in motion can be expressed as E(m) =
, where p is the momentum
and m is the mass of the object. If sand is added to a moving railway car, what would happen as m continues to
increase?
SOLUTION: Sample answer: As the mass of the railway car continues to increase, the railway car’s kinetic energy will approach
0.
Use each graph to determine the x-value(s) at which each function is discontinuous. Identify the type of
discontinuity. Then use the graph to describe its end behavior. Justify your answers.
43. SOLUTION: h(x) has an infinite discontinuity at x = 0 because h(0) is undefined and h(x) approaches
the left and the right.
x
0.001
0.01
0.1
−0.1 −0.01 −0.001 0
h(x)
136.4 13,636 1.36 · 106 1.36 · 106 13,636 136.4
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From the graph, it appears that as x → –
and
, h(x)
0.
as x approaches 0 from
Page 1
increase?
SOLUTION: answer: End
As theBehavior,
mass of the railway
car continues to increase, the railway car’s kinetic energy will approach
1-3 Sample
Continuity,
and Limits
0.
Use each graph to determine the x-value(s) at which each function is discontinuous. Identify the type of
discontinuity. Then use the graph to describe its end behavior. Justify your answers.
43. SOLUTION: h(x) has an infinite discontinuity at x = 0 because h(0) is undefined and h(x) approaches
the left and the right.
x
0.001
0.01
0.1
−0.1 −0.01 −0.001 0
h(x)
136.4 13,636 1.36 · 106 1.36 · 106 13,636 136.4
From the graph, it appears that as x → –
and
, h(x)
as x approaches 0 from
0.
The table supports this conjecture.
GRAPHING CALCULATOR Graph each function and determine whether it is continuous. If
discontinuous, identify and describe any points of discontinuity. Then describe its end behavior and
locate any zeros.
45. f (x) =
SOLUTION: Use the TABLE and TRACE functions on the calculator to locate points of discontinuity and zeros. The graph has
infinite discontinuities at x = −1, x = 2, and x = 3.
There is a zero at x = 0.
47. h(x) =
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Page 2
1-3
Use the TABLE and TRACE functions on the calculator to locate points of discontinuity and zeros. The graph has
infinite
discontinuities
x = −1, x = 2,and
and xLimits
= 3.
There is a zero at x = 0.
Continuity,
EndatBehavior,
47. h(x) =
SOLUTION: Use the TABLE and TRACE functions on the calculator to locate points of discontinuity and zeros. The graph has
infinite discontinuities at x = 3 and x = −6.
There are zeros at x = −3 and x =
.
49. h(x) =
SOLUTION: Use the TABLE and TRACE functions on the calculator to locate points of discontinuity and zeros. The graph has
infinite discontinuities at x = −4 and x = 3.
There are zeros at x = −5, x = 4, and x =
6.
GRAPHING CALCULATOR Graph each function, and describe its end behavior. Support the
conjecture numerically, and provide an effective viewing window for each graph.
51. f (x) = −x4 + 12x3 + 4x2 – 4
SOLUTION: end behavior:
= −
,
= −
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Use the TABLE and TRACE functions on the calculator to locate points of discontinuity and zeros. The graph has
infinite discontinuities at x = −4 and x = 3.
There are zeros at x = −5, x = 4, and x =
1-3 Continuity, End Behavior, and Limits
6.
GRAPHING CALCULATOR Graph each function, and describe its end behavior. Support the
conjecture numerically, and provide an effective viewing window for each graph.
51. f (x) = −x4 + 12x3 + 4x2 – 4
SOLUTION: end behavior:
= −
,
= −
53. f (x) =
SOLUTION: end behavior:
= 16, eSolutions Manual - Powered by Cognero
= 16
Page 4
55. BUSINESS Gabriel is starting a small business screen-printing and selling T-shirts. Each shirt costs $3 to produce.
1-3 Continuity, End Behavior, and Limits
53. f (x) =
SOLUTION: end behavior:
= 16, = 16
55. BUSINESS Gabriel is starting a small business screen-printing and selling T-shirts. Each shirt costs $3 to produce.
He initially invested $4000 for a screen printer and other business needs.
a. Write a function to represent the average cost per shirt as a function of the number of shirts sold n.
b. Use a graphing calculator to graph the function.
c. As the number of shirts sold increases, what value does the average cost approach?
SOLUTION: a.
b.
c. $3; Sample answer: As n approaches
number of shirts sold increases.
, f (n) approaches 3. Therefore, the average cost approaches $3 as the
57. GRAPHING CALCULATOR Graph several different functions of the form f (x) = xn + axn − 1 + bxn − 2, where
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n, a, and b are nonnegative integers.
a. Make a conjecture about the end behavior of the function when n is positive and even. Include a graph to support
your conjecture.
$3; Sample answer:
As n approaches
, f (n) approaches 3. Therefore, the average cost approaches $3 as the
1-3 c.
Continuity,
End Behavior,
and Limits
number of shirts sold increases.
57. GRAPHING CALCULATOR Graph several different functions of the form f (x) = xn + axn − 1 + bxn − 2, where
n, a, and b are nonnegative integers.
a. Make a conjecture about the end behavior of the function when n is positive and even. Include a graph to support
your conjecture.
b. Make a conjecture about the end behavior of the function when n is positive and odd. Include a graph to support
your conjecture.
SOLUTION: a. If n is even, f (x) approaches
as x approaches –
b. Sample answer: If n is odd, f(x) approaches –
and
.
as x approaches –
and
as x approaches
.
REASONING Determine whether each function has an infinite, jump , or removable discontinuity at x = 0.
Explain.
59. f (x) =
SOLUTION: Infinite; f (0) is undefined, and f (x) approaches –
the right.
as x approaches 0 from the left and
as x approaches 0 from
61. CHALLENGE Determine the values of a and b so that f is continuous.
SOLUTION: 2
When x = 3, x + a must equal bx + a. When x = −3, bx + a must equal
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.
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SOLUTION: Infinite; f (0) is undefined, and f (x) approaches –
as x approaches 0 from the left and
1-3 the
Continuity,
End Behavior, and Limits
right.
as x approaches 0 from
61. CHALLENGE Determine the values of a and b so that f is continuous.
SOLUTION: 2
When x = 3, x + a must equal bx + a. When x = −3, bx + a must equal
.
GRAPHING CALCULATOR Graph each function. Analyze the graph to determine whether each function is even, odd, or neither. Confirm algebraically. If odd or even, describe the symmetry of the
graph of the function.
68. f (x) =
SOLUTION: The function is neither even nor odd.
State the domain of each function.
71. g(x) =
SOLUTION: The domain is restricted to when the denominator does not equal 0.
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1-3 The
Continuity,
End Behavior,
function is neither
even nor odd. and Limits
State the domain of each function.
71. g(x) =
SOLUTION: The domain is restricted to when the denominator does not equal 0.
Therefore, x cannot equal 1 −
or 1 +
.
73. POSTAL SERVICE The U.S. Postal Service uses five-digit ZIP codes to route letters and packages to their
destinations.
a. How many ZIP codes are possible if the numbers 0 through 9 are used for each of the five digits?
b. Suppose that when the first digit is 0, the second, third, and fourth digits cannot be 0. How many five-digit ZIP
codes are possible if the first digit is 0?
c. In 1983, the U.S. Postal Service introduced the ZIP + 4, which added four more digits to the existing five-digit
ZIP codes. Using the numbers 0 through 9, how many additional ZIP codes were possible?
SOLUTION: a. There are 10 possibilities for each digit.
10 · 10 · 10 · 10 · 10 = 100,000
b. There is only 1 possibility for the first digit, 10 possibilities for the 5th digit, and 9 possibilities for the other digits.
1 · 9 · 9 · 9 · 10 = 7290
c. 1,000,000,000 − 100,000 = 999,900,000
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