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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
142 Chapter 1 • Limits and Continuity
83. (a) 0.26 moles
(b) 395 min
(c) 0 moles
(d) In the long run, all of the sucrose
will decompose. See TSM.
84. (a) Not continuous.
(b) A camera cannot focus on an object
placed close to its focal length because
the distance of the image from the lens
becomes unbounded.
85. a
≠ 0; b can be any real number;
c = 0; and d ≠ 0. See TSM.
86. a ≠ 0; b can be any real number;
c = 0; and d ≠ 0.
87. See TSM.
88. See TSM.
89. See TSM.
90. See TSM.
91. (a) For table, see TSM.
(b) e ≈ 2.718281828
(c) Answers will vary. Sample answer:
The results from (a) and (b) agree to
five decimal places.
92. The property requires the exponent to
be a constant, independent of x, but in
x
⎛ 1 ⎞
lim +
→∞⎝
⎜1
x ⎠
⎟ the exponent is x.
x
93. (a) ∞
(b) The result suggests that it is not
possible to reach the speed of light.
lens to the object being photographed and q is the distance from
the lens to the image formed by the lens. See the figure below.
To photograph an object, the object’s image must be formed on
the photo sensors of the camera, which can only occur if q is
positive.
Object
p
Lens
q
Photo
sensors
(a) Is the distance q of the image from the lens continuous as the
distance of the object being photographed approaches the
focal length f of the lens? (Hint: First solve the thin-lens
equation for q and then find lim q.) p→ f +
(b) Use the result from (a) to explain why a camera (or any lens)
cannot focus on an object placed close to its focal length.
In Problems 85 and 86, find conditions on a, b, c, and d so that the
graph of f has no horizontal or vertical asymptotes.
85. f (x) = ax3 + b
cx 4 + d
86. f (x) = ax + b
cx + d
87. Explain why the following properties are true. Give an example of
each.
1
(a) If n is an even positive integer, then lim x→c (x − c) n =∞.
1
(b) If n is an odd positive integer, then lim
x→c − (x − c) n = −∞.
1
(c) If n is an odd positive integer, then lim
x→c + (x − c) n =∞.
88. Explain why a rational function, whose numerator and
denominator have no common zeros, will have vertical
asymptotes at each point of discontinuity.
89. Explain why a polynomial function of degree 1 or higher cannot
have any asymptotes.
AP® Practice Problems
PAGE
138 1. For x > 0, the line y = 1 is an asymptote of the graph of a
function f . Which of the following statements must be true?
(A) f (x) = 1 for x > 0. (B) lim f (x) =∞
x→1
(C)
lim
x→∞
PAGE
3x 3 + 4x 2 − x + 10
135 2. lim
x→∞ 2x 4 − x 3 + 2x 2 − 2 =
3
(A) –5 (B) 0 (C)
2
PAGE
5x 3 − x
135 3. lim
x→∞ 8 − x 3 =
5
(A) –5 (B) (C) 5
8
f (x) = 1 (D) lim
x→−∞ f (x) = 1
(D) ∞
(D) ∞
90. If P and Q are polynomials of degree m and n, respectively,
P(x)
discuss lim
x→∞ Q(x) when:
(a) m > n (b) m = n (c) m < n
91. (a) Use a table to investigate lim 1 + 1 x
.
x→∞ x
CAS (b) Find lim 1 + 1 x
.
x→∞ x
(c) Compare the results from (a) and (b). Explain the possible
causes of any discrepancy.
Challenge Problems
92. lim 1 + 1
x→∞ x
= 1, but lim
x→∞
1 + 1 x
> 1. Discuss why the
x
lim
x→∞ f (x) n
cannot be used to find the
property lim [ f x→∞ (x)]n =
second limit.
93. Kinetic Energy At low speeds the kinetic energy K , that is, the
energy due to the motion of an object of mass m and speed v, is
given by the formula K = K (v) = 1 2 mv2 . But this formula is
only an approximation to the general formula, and works only for
speeds much less than the speed of light, c. The general formula,
which holds for all speeds, is
⎡
⎤
Kgen(v) = mc 2 ⎢
1
⎣
− 1⎥
⎦
1 − v2
c 2
(a) As an object is accelerated closer and closer to the speed of
light, what does its kinetic energy Kgen approach?
(b) What does the result suggest about the possibility of reaching
the speed of light?
PAGE
138 4. Find all the horizontal asymptotes of the graph of y = 2 + 3x
4 − 3 x .
(A) y = –1 only
(B) y = 1 2 only
(C) y = –1 and y = 0 (D) y = –1 and y = 1 2
PAGE
131 5. Find all the vertical asymptotes of the graph of
r(x) = x2 + 5x + 6
.
x 3 − 4x
(A) x = 0 and x = –2 (B) x = 0 and x = 2
(C) x = –2 and x = 2 (D) x = 0, x = –2 and x = 2
PAGE
134 6. lim
x→−∞
√
8x 2 − 4x
x + 2
=
(A) –∞ (B) −2 √ 2 (C) 4 (D) 2 √ 2
Answers to AP® Practice Problems
1. C
2. B
3. A
4. D
5. B
6. B
142
Chapter 1 • Limits and Continuity
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