1_ Limits and Continuity WITH ANSWERS

AP Calculus AB Exam Review
Limits and Continuity
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) Assume that a watermelon dropped from a tall building falls y = 16t2 ft in t sec. Find the
watermelon's average speed during the first 6 sec of fall.
A) 97 ft/sec
B) 48 ft/sec
C) 96 ft/sec
1)
D) 192 ft/sec
Determine the limit by substitution.
2)
lim
x→4
x2 + 14x + 49
2)
A) 121
B) Does not exist
C) ± 11
D) 11
Determine the limit algebraically, if it exists.
x3 + 12x2 - 5x
3) lim
5x
x→0
A) 0
3)
B) Does not exist
C) -1
D) 5
Determine the limit graphically, if it exists.
4)
lim f(x)
x→1 -
4)
5
f(x)
4
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
5 x
-1
-2
-3
-4
-5
A) Does not exist
B)
1
2
C) -1
D) 2
Find the indicated limit.
5)
lim int x
x→-3A) -3
5)
B) -4
C) 4
1
D) 0
Match the function with the correct table values.
x2 - 2x - 8
6) f(x) =
x2 - 9x + 20
6)
x 3.9 3.99 3.999 4.001 4.01 4.1
f(x)
A) -5.2636; -5.8307; -5.8930; -5.9070; -5.9707; -6.6778
B) -5.4636; -6.0307; -6.0930; -6.1070; -6.1707; -6.8778
C) 0.2308; 0.2231; 0.2223; 0.2221; 0.2214; 0.2135
D) -5.3636; -5.9307; -5.9930; -6.0070; -6.0707; -6.7778
Find the limit.
7) Let
lim f(x) = 49. Find lim
x → -1
x → -1
A) -1
f(x).
7)
B) 7
C) 2.6458
Evaluate or determine that the limit does not exist for each of the limits (a)
D) 49
lim f(x), (b) lim f(x), and (c) lim f(x)
x→dx→d+
x→d
for the given function f and number d.
8)
8)
f(x) =
x2 - 4,
3,
for x < 0,
for x ≥ 0
d = -4
A) (a) -4
(b) 3
(c) Does not exist
B) (a) 12
(b) 12
(c) 12
C) (a) -4
(b) 3
(c) 3
D) (a) 3
(b) -4
(c) Does not exist
Provide an appropriate response.
9) It can be shown that the inequalities
1
-x ≤ x cos( ) ≤ x
x
9)
1
hold for all values of x ≥ 0. Find lim x cos( ) if it exists
x
x→0
A) Does not exist
B) 0.0007
C) 1
D) 0
Find the limit, if it exists.
cos 3x
10) lim
x
x→-∞
A) 0
10)
B) -∞
C) 3
2
D) 1
Find the indicated limit.
sin x
11) lim
x→-∞ x
A) 1
11)
B) ∞
C) 0
D) Does not exist
Find the limit.
12)
lim (1 + csc x )
x→0 +
A) -∞
12)
B) 1
C) 0
D) ∞
Find the vertical asymptotes of the graph of f(x).
13) f(x) = sec x
π
A) x = + nπ, n is any integer
2
13)
B) x = 0
C) x = nπ, n is any integer
D) x = π
3
Match the function with the graph of its end behavior model.
2x3 - 5x2 + 1
14) y =
x+9
A)
14)
B)
y
-40
y
400
400
200
200
-20
20
40
x
-40
-20
-200
-200
-400
-400
C)
20
40
x
20
40
x
D)
y
y
-40
400
400
200
200
-20
20
40
x
-40
-20
-200
-200
-400
-400
Find a power function end behavior model.
15) y = 4x2 - 2x + 7
A) y = 4x2
15)
B) y = 4x2 + 7
C) y = 4x2 - 2x
D) y = 4x - 2
Find a simple basic function as a right-end behavior model and a simple basic function as a left-end behavior model.
16) y = 4x + ln x
16)
A) y = 4x; y = -4x
B) y = 4x; y = 4x
C) y = 4x; y = ln x
D) y = x; y = -ln x
Find the limit, if it exists.
17)
lim x4ex
x→-∞
A) 0
17)
B) 4
C) 1
4
D) ∞
Find the limit of f(x) as (a) x→-∞, (b) x→∞, (c) x→0-, and (d) x→0+.
x-4
, x≤0
x-2
18) f(x) =
1
,
x2
18)
x>0
A) (a) -∞
(b) ∞
(c) 4
(d) 2
B) (a) 1
(b) 0
(c) 2
(d) ∞
C) (a) 2
(b) Does not exist
(c) 1
(d) 0
D) (a) ∞
(b) 2
(c) 0
(d) 1
Find the intervals on which the function is continuous.
4
19) y = 9x - 8
8
8
A) - , ∞
B) -∞,
9
9
19)
8
C) , ∞
9
8
D) , ∞
9
Find the points of discontinuity. Identify each type of discontinuity.
x+4
20) y =
2
x - 14x + 48
20)
A) x = 6, x = 8, both infinite discontinuities
B) x = 6, infinite discontinuity
C) x = -6, x = -8, both infinite discontinuities
D) x = 8, infinite discontinuity
Find all points where the function is discontinuous.
21)
21)
A) x = 4
B) None
C) x = 2
D) x = 4, x = 2
Give a formula for the extended function that is continuous at the given point.
sin 5x
22) f(x) =
,x=0
x
A) y =
sin 5x
, x≠0
x
5,
B) y =
x=0
sin 5x
, x=0
x
5,
C) y = 5x
D) y = sin 5x
5
22)
x≠0
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
23) Verify that the function f(x) =
-4x2 + x
is continuous. Indicate which theorems are needed
x
23)
and which functions are assumed to be continuous for all x in the domain.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find a value for a so that the function f(x) is continuous.
2
24) f(x) = x - 5, x < 4
5ax,
x≥4
A) a = 9
24)
B) a = 11
C) a =
11
20
D) a =
4
5
Solve the problem.
25) Consider the learning curve defined in the graph. Depicted is the accuracy, p, expressed as a
percentage, in performing a series of short tasks versus the accumulated amount of time spent
practicing the tasks, t. Is p(t) continuous at t = 25? at t = 40? at t = 45?
A) Yes; no; no
B) No; no; no
C) Yes; no; yes
6
D) Yes; yes; yes
25)
Testname: LIMITS AND CONTINUITY
1)
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23)
C
D
C
D
B
D
B
B
D
A
C
D
A
D
A
B
A
B
C
A
A
A
Assume y = x and the square root function are continuous. Use the sum, constant multiple, product, and quotient
theorems of continuity.
24) C
25) C
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