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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.5 • Assess Your Understanding 139
3
y
2
y
8
4
8 4
4 8 12
4
8
12
5
x
2
Figure 64 R(x) = 3x2 − 12
2x 2 − 9x + 10
1.5 Assess Your Understanding
Concepts and Vocabulary
1. True or False ∞ is a number.
1
2. (a) lim =
x→0 − x
1
; (b) lim
x→0 + x = ;
(c) lim x→0 +
x
To determine the behavior to the right of x = 5 , we find the right-hand limit.
2
lim
x→ 5 +
2
R(x) = lim
x→ 5 +
2
[ ] 3(x + 2)
(2x − 5)
= 3 lim
x→ 5 +
2
x + 2
2x − 5 =∞
As x approaches 5 from the right, R becomes unbounded in the positive direction. The
2
graph of R has a vertical asymptote on the right at x = 5 2 .
Next we consider lim
x→2
R(x).
lim R(x) = lim
x→2 x→2
3. True or False The graph of a rational function has a vertical
asymptote at every number x at which the function is not defined.
4. If lim f (x) =∞, then the line x = 4 is a(n)
x→4
asymptote
of the graph of f.
1
1
5. (a) lim = ; (b) lim
x→∞ x x→∞ x 2 = ;
(c) lim x→∞
6. True or False lim x→−∞
7. (a) lim
x→−∞ ex = ; (b) lim
x→∞ ex = ; (c) lim
x→∞ e−x =
8. True or False The graph of a function can have at most two
horizontal asymptotes.
3(x − 2)(x + 2)
(2x − 5)(x − 2) = lim
x→2
3(x + 2)
2x − 5
=
3(2 + 2)
2 · 2 − 5 = 12
−1 =−12
Since the limit is not infinite, the function R does not have a vertical asymptote at 2.
Since 2 is not in the domain of R, the graph of R has a hole at the point (2, −12).
To check for horizontal asymptotes, we find the limits at infinity.
3x 2 − 12
lim R(x) = lim
x→∞ x→∞ 2x 2 − 9x + 10 = lim 3x 2
x→∞ 2x = lim 3
2 x→∞ 2 = 3 ↑
2
(2)
lim R(x) = lim 3x 2 − 12
x→−∞ x→−∞ 2x 2 − 9x + 10 = lim 3x 2
x→−∞ 2x = lim 3
2 x→−∞ 2 = 3 ↑
2
(2)
The line y = 3 is a horizontal asymptote of the graph of R for x unbounded in the
2
negative direction and for x unbounded in the positive direction. ■
The graph of R and its asymptotes are shown in Figure 64. Notice the hole in the
graph at the point (2, −12).
NOW WORK Problem 69 and AP® Practice Problems 8 and 10.
Skill Building
In Problems 9–16, use the accompanying graph of y = f (x).
9. Find lim f (x).
x→∞
10. Find lim f (x).
x→−∞
PAGE
129 11. Find lim f (x).
x→−1 −
12. Find lim f (x).
x→−1 +
13. Find lim f (x).
x→3 −
14. Find lim f (x).
x→3 +
PAGE
131 15. Identify all
vertical
asymptotes.
16. Identify all
horizontal
asymptotes.
y
12
8
4
4 4 8 12
x 1 x 3
y 2
x
Must-Do Exercises for
Exam Readiness
AB: 2–26, 27–59 odd, 67–71 odd, AP ®
Practice Problems
BC: 17–26, 30–35, 45, 53, 55, 63, 73, 78,
all AP ® Practice Problems
TRM Full Solutions to Section
1.5 Problems and AP® Practice
Problems
Answers to Section 1.5
Problems
1. False.
2. (a) −∞ (b) ∞ (c) − ∞
3. False.
4. Vertical.
5. (a) 0 (b) 0 (c) ∞
6. False.
7. (a) 0 (b) ∞ (c) 0
8. True.
9. 2
10. 0
11. ∞
12. ∞
13. ∞
14. ∞
15. x = −1, x = 3
16. y = 0, y = 2
Section 1.5 • Assess Your Understanding
139
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