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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Section 1.4 • Assess Your Understanding 125
Figures 47(a) and 47(b) illustrate the graphs of f and F.
y
8
4
y
1
1
2
4
x
Must-Do Problems for
Exam Readiness
AB: 9–45 odd, AP ® Practice Problems
BC: 11, 23, 25, 35, 43, 45, 55, all AP ®
Practice Problems
Summary Basic Limits
• lim
x→c
A = A, where A is a constant
• lim sin x = sin c • lim cos x = cos c
x→c x→c
• lim a x = a c , a > 0, a = 1
x→c
1.4 Assess Your Understanding
Concepts and Vocabulary
1. lim x→0
sin x =
Figure 47
• lim
x→c
x = c
2 2
(a) f (x) e 2x
sin θ
cos θ − 1
• lim = 1 • lim = 0
θ→0 θ
θ→0 θ
• lim
x→c
log a x = log a c, c > 0, a > 0, a = 1
cos x − 1
2. True or False lim = 1
x→0 x
3. The Squeeze Theorem states that if the functions f, g, and h
have the property f (x) ≤ g(x) ≤ h(x) for all x in
an open interval containing c, except possibly at c, and
if lim f (x) = lim h(x) = L, then lim g(x) = .
x→c x→c x→c
4. True or False f (x) = csc x is continuous for all real numbers
except x = 0.
Skill Building
In Problems 5–8, use the Squeeze Theorem to find each limit.
PAGE
118 5. Suppose −x 2 + 1 ≤ g(x) ≤ x 2 + 1 for all x in an open
interval containing 0. Find lim g(x).
x→0
6. Suppose −(x − 2) 2 − 3 ≤ g(x) ≤ (x − 2) 2 − 3 for all x
in an open interval containing 2. Find lim x→2
g(x).
7. Suppose cos x ≤ g(x) ≤ 1 for all x in an open interval
containing 0. Find lim x→0
g(x).
8. Suppose −x 2 + 1 ≤ g(x) ≤ sec x for all x in an open interval
containing 0. Find lim x→0
g(x).
In Problems 9–22, find each limit.
9. lim x→0
(x 3 + sin x) 10. lim x→0
(x 2 − cos x)
x
2
3
(b) F(x) ln x
NOW WORK Problem 45 and AP® Practice Problems 3, 5, and 9.
11. lim (cos x + sin x) 12. lim (sin x − cos x)
x→π/3 x→π/3
13.
cos x
sin x
lim
14. lim
x→0 1 + sin x
x→0 1 + cos x
15.
3
e x − 1
lim x→0 1 + e x 16. lim x→0 1 + e x
17. lim(e x sin x) 18. lim(e −x tan x)
x→0 x→0
( ) e x
( x
)
19. lim ln
20. lim ln
x→1 x
x→1 e x
e 2x
1 − e x
21. lim x→0 1 + e x 22. lim x→0 1 − e 2x
In Problems 23–34, find each limit.
PAGE
121 23.
sin(7x)
lim x→0 x
PAGE
121 25.
θ + 3 sin θ
lim θ→0 2θ
27.
sin θ
lim θ→0 θ + tan θ
29.
5
lim θ→0 θ · csc θ
PAGE
122 31.
1 − cos 2 θ
lim θ→0 θ
24. lim x→0
sin x 3
x
26.
2x − 5 sin(3x)
lim x→0 x
28.
tan θ
lim θ→0 θ
30.
sin(3θ)
lim θ→0 sin(2θ)
[
33. lim(θ · cot θ) 34. lim sin θ
θ→0 θ→0
cos(4θ)− 1
32. lim θ→0 2θ
( )]
cot θ − csc θ
θ
TRM Full Solutions to Section
1.4 Problems and AP® Practice
Problems
Answers to Section 1.4
Problems
1. 0
2. False.
3. L
4. False.
5. 1
6. −3
7. 1
8. 1
9. 0
10. −1
3 1
11. +
2 2
3 1
12. −
2 2
13. 1
14. 0
15. 3 2
16. 0
17. 0
18. 0
19. 1
20. −1
21. 1 2
TRM Section 1.4: Worksheet 2
This worksheet includes 9 limit questions to be
solved analytically.
22. 1 2
23. 7
24. 1 3
25. 2
26. −13
27. 1 2
28. 1
29. 5
30. 3 2
31. 0
32. 0
33. 1
34. 0
Section 1.4 • Assess Your Understanding
125
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