### 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)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
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
7
```