Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
The solution of x3–8x2> 0 is
1)
A)
x< 0, x> 8.
B)
x> 0.
C)
x< 0.
D)
x> 8.
E)
0 <x< 8.
2)
The solution of (x+ 1)(x2– 1) < 0 is
2)
A)
x< 1.
B)
x< – 1, x> 1.
C)
–1 <x< 1.
D)
x> 1.
E)
x< – 1, –1 <x< 1.
3)
lim
x
1–
7
1 – x =
3)
A)
0
B)
7
C)
D)
–
E)
7
2
1
4)
The solution of x2– 5x– 14 < 0 is
4)
A)
x< – 2, x> 7
B)
x> 7
C)
x< – 2
D)
–2 <x< 7
E)
all real numbers
5)
lim
h
0
x2+ h
x + 2h=
5)
A)
B)
–
C)
0
D)
x
E)
does not exist
D
6)
If f(x) =x + 2, if x> 0
x2+ 3, if x < 0 , then lim
x
0–
f(x) =
6)
A)
B)
2
C)
3
D)
0
E)
does not exist
C
2
D
7)
The solution of x2– 4
x< 0 is
7)
A)
–2 <x< 0, 0 <x< 2.
B)
–2 <x< 0, x> 2.
C)
–2 <x< 0, x> 0.
D)
x< – 2, 0 <x< 2.
8)
If f(x) =2x,if x> 1
2, if x< 1 , then lim
x
1f(x) =
8)
A)
0
B)
1
C)
2
D)
E)
does not exist
C
9)
lim
x
2+
4
x– 2 =
9)
A)
4
B)
–2
C)
D)
–
E)
0
C
3
E)
–2 <x< 0, 0 <x< 2, x> 2.
D
10)
lim
x
3 =
10)
A)
B)
3
C)
–
D)
0
E)
does not exist
11)
lim
x
1
2x2+ x – 3
x2+4x – 5
=
11)
A)
5
6
B)
2
5
C)
3
5
D)
0
E)
1
4
A
12)
If f(x) =x + 1, if x 1
x– 1, if x < 1 , then lim
x
1f(x) =
12)
A)
2
B)
–
C)
0
D)
E)
does not exist
E
13)
lim
x–3
x2+2x– 3
x2+ 7x + 12
=
13)
A)
0
B)
–
C)
–4
D)
4
E)
2
C
4
B
14)
The solution of 9x2+ 6x+ 1 < 0 is
14)
A)
x> – 1
3.
B)
x< – 1
3.
C)
–1
3<x<1
3.
D)
all real numbers
E)
no solution
15)
lim
x
3+
3x– 9 =
15)
A)
B)
3
C)
9
D)
0
E)
1
B)
E)
16)
lim
x–
4 –x2
1 –x=
16)
A)
0
B)
–
C)
–4
D)
E)
4
B)
E)
17)
The solution of x2+ 2x– 15
0 is
17)
A)
–5 x
3.
B)
x–5.
C)
x 3.
D)
x–5.
E)
x–5, x
3.
B)
E)
5
B)
E)
18)
The solution of (x+ 2)(x+ 4)(2x– 3) < 0 is
18)
A)
x< – 4, –2 <x<3
2.
B)
–2 <x<3
2.
C)
–4 <x< – 2.
D)
–4 <x< – 2, x>3
2.
E)
x< – 4, x>3
2.
19)
lim
x–5
x+ 3
5 –x=
19)
A)
0
B)
C)
–1
5
D)
–
E)
4
5
20)
lim
x–1
x2+4x+ 3
x2– x – 2
=
20)
A)
–2
3
B)
C)
0
D)
–4
E)
1
6
21)
If f(x) =3, if x> 4
1, if x 1 , then lim
x
4+
f(x) =
21)
A)
0
B)
1
C)
3
D)
4
E)
does not exist
22)
lim
x
4x2+2x+ 1
(x– 1)2=
22)
A)
4
B)
C)
0
D)
1
E)
2
A
23)
The solution of x+ 5
x2+ 2x– 8
0 is
23)
A)
–5 x< – 4, x> 2.
B)
x–4, x > 2.
C)
x= – 5, x> 2.
D)
x–5, –4 x 2.
E)
x–5, –4 <x< 2.
A
24)
lim
x–
4 +x2
x2+x
=
24)
A)
0
B)
4
C)
D)
–
E)
1
E
7
C
25)
The solution of (x+ 9)(x– 1)(x– 5) 0 is
25)
A)
x 5.
B)
x–9.
C)
–9 x
1, x
5.
D)
–9 x
1.
E)
x–9, 1 x
5.
26)
The solution of x2– 1
x2– 5x+ 6
> 0 is
26)
A)
x< – 1, 1 <x< 3.
B)
1 <x< 2, x> 3.
C)
x< – 1, 1 <x< 2, x> 3.
D)
x< 2, 2 <x3, x > 3.
E)
–1 <x< 1, 2 <x< 3.
27)
lim
q
2
4q2– q + 2
8=
27)
A)
2
B)
5
C)
1
4
D)
4
E)
3
28)
lim
x
0–
x2+ 4
x=
28)
A)
4
B)
–
C)
D)
0
E)
1
8
29)
If f(x) = 3x– 7, then lim
h
0
f(x+h) –f(x)
h=
29)
A)
3x.
B)
0.
C)
–7.
D)
3.
E)
does not exist
30)
Let f(x) =4
x2+ 9 . The only value(s) of x for which f is discontinuous is (are)
30)
A)
3.
B)
–3 and 3.
C)
4.
D)
–3.
E)
none
E
31)
lim
x–
x– 1
4x2– 5
=
31)
A)
–
B)
C)
0
D)
1
4
E)
1
5
C
32)
lim
x–2(2 +3)=
32)
A)
2 +3
B)
0
C)
–2 +3
D)
3
E)
does not exist
A
9
D
33)
Let f(x) =x2– 9
x2+ 2x+ 1 . The only value(s) of x for which f is discontinuous is (are)
33)
A)
1, –3 and 3.
B)
–1, 1, –3, and 3.
C)
–1 and 1.
D)
–1.
E)
–3 and 3.
E)
34)
lim
x
5 –x2
(x4– 8x2+ 2)
=
34)
A)
5
2
B)
0
C)
D)
–
E)
5
B)
E)
35)
lim
x
2x2+x+ 1
=
35)
A)
3
2
B)
C)
–
D)
–4
E)
2
B)
E)
B)
36)
Let f(x) =x(x+ 1)
x2– 1 . The only value(s) of x for which f is discontinuous is (are)
36)
A)
–1.
B)
0.
C)
–1 and 1.
D)
–1, 0, and 1.
E)
1.
37)
The solution of 2x– 5
x+ 3 0 is
37)
A)
x–3, x>5
2.
B)
–3 x5
2.
C)
x< – 3, x5
2.
D)
x–3, x5
2.
E)
–3 <x5
2.
38)
lim
t
2
t2– t – 2
t2+3t – 10
=
38)
A)
3
7
B)
1
5
C)
D)
–1
E)
–
39)
The solution of 2x2– 13x+ 6 0 is
39)
A)
x 6.
B)
1
2x
6.
C)
x1
2.
D)
x1
2, x 6.
E)
no solution
40)
If f(x) =x,if x 2
2 –x,if x < 2 , then lim
x
2f(x) =
40)
A)
0
B)
1
C)
2
D)
E)
does not exist
E
41)
lim
q
2q+ 3 =
41)
A)
0
B)
D
–1
C)
–
D)
E)
7
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
42)
Find: lim
x
5
x2
h
42)
12
B
43)
Suppose that the rent for an apartment in a city with rent control is $540 per month. Show
that the rent function R(t) = 540 is continuous at t= 3.
43)
44)
Find: lim
x–2
x2+ 4x+ 4
x2– 4
44)
45)
The profit function for a product is given by P(x) = 12x+x2– 4320. Determine for what
values of x the profit is positive.
45)
46)
Find: lim x(x– 1)
x
3
46)
47)
Find: lim
x
6
x3–6x2
x– 6
47)
48)
The volume of helium in a spherical balloon (in cubic centimeters) as a function of the
radius, r, in centimeters, is given by V(r) =4
3r3. Find lim
r
1
V(r).
48)
49)
If f(x) =4, if x>3
x+ 1, if x< 3 , find lim
x
3f(x). Hint: Sketch the graph of f.
49)
50)
Solve the inequality: x2– 1
x2– 4
< 0.
50)
51)
The cost of purifying water is given by C=18,000
p– 2000 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Find lim
p
0C.
51)
52)
Find: lim
x
1+
4x– 1
3x. If the limit does not exist, so state or use the symbol or – if
appropriate.
52)
53)
Solve: (x– 1)(x+ 2)(x– 3) 0
53)
54)
The greatest integer function in mathematics (denoted f(x) =x) is used every day by
cashiers making change for customers. This function tells the amount of paper money for
each amount of change owed. (For example, if the customer is owed $1.25 in change, he
would get $1 in paper money, thus 1.25 = 1). By considering 0.90 , 0.95 , 0.99 , 1.01 ,
and 1.05 , determine lim
x
1x.
54)
55)
Find the value(s) of x for which f(x) =x2,if x
2
3x,if x< 2 is discontinuous.
55)
56)
Find lim
h
0
f(x+h) –f(x)
h where f(x) =2x2+ 3x+ 5
56)
57)
The revenue function for a certain product is given by R(x) = 500x–6x2. Find lim
x
4R(x).
57)
58)
The length of a material increases as it is heated up according to the equation l= 125 + 5x.
The rate at which the length is increasing is given by: lim
h
0
125 + 5(x+h) – (125 + 5x)
h.
Calculate this limit.
58)
14
59)
Find the value(s) of x for which f(x) =x– 2
x2+ 4x is discontinuous.
59)
60)
Use your calculator to complete the table, and use your results to estimate the given limit.
lim
x
1
2
2x2 + 9x– 5
2x– 1
x0.51 0.501 0.5001 0.4999 0.499 0.49
f(x)
60)
61)
Find: lim
x–4
3 –x
x – 4
61)
62)
Find: lim
x
4x2– 6x
x2+ 4x. If the limit does not exist, so state or use the symbol or – if
appropriate.
62)
63)
The yearly sales y of a certain company (in thousands of dollars) is related to the amount
that company spends on advertising, x (in thousands of dollars), according to the equation
y(x) =400x
x+ 25 . Find lim
x
y(x) and discuss what this means to the company.
63)
64)
Find lim
h
0
f(x+h) –f(x)
h where f(x) =1
2x+ 3
64)
65)
Find the value(s) of x for which f(x) =1
1+ x2 is discontinuous.
65)
Answer:
Explanation:
66)
Find: lim
x
3+
x2– 6
x+ 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
66)
Answer:
Explanation:
67)
The cost of purifying water is given by C=8000
p– 5500 where C is the cost in dollars and p
is the percent of impurities left in the water after purification. Find lim
p
0C.
67)
Answer:
Explanation:
68)
Solve: x– 1
x– 4 0
68)
Answer:
x 1, x> 4
Explanation:
69)
The length of a material increases as it is heated up according to the equation l= 140 + 0.8x.
The rate at which the length is increasing is given by: lim
h
0
140 + 0.8(x+h) – (140 + 0.8x)
h.
Calculate this limit.
69)
Answer:
Explanation:
16
Answer:
Explanation:
70)
Consider the graph of f(x)
Determine the following limits:
(a) lim
x
3+f(x)(d) lim
x
2+
f(x)(g) lim
x
f(x)
(b) lim
x
3–
f(x)(e) lim
x
2–
f(x)(h) lim
x
f(x)
(c) lim
x
3f(x)(f) lim
x
2f(x)
70)
71)
Solve: x2–x– 6
0
71)
72)
The revenue function for a certain product is given by R(x) = 500x–6x2. Find lim
x
0R(x).
72)
73)
Find the value(s) of x for which f(x) =
1
x,if x> 1
3x,if x< 1
is discontinuous.
73)
74)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
1+
x– 1
x2– 2x+ 1
74)
75)
The yearly sales y of a certain company (in thousands of dollars) is related to the amount
that company spends on advertising, x (in thousands of dollars), according to the equation
y(x) =1000x
x+ 50 . Graph this function on your graphing calculator in the window 0,100 ×
0,1100 . Use TRACE to explore lim
x
y(x) and discuss what this means to the company.
75)
76)
Find lim
h
0
f(x+h) –f(x)
h where f(x) =2x+ 3
76)
77)
Solve: (x–1)2(x+ 4) > 0
77)
78)
Solve: x2– 6x+ 8
x2– 1
< 0
78)
18
79)
Let f(x) =
x,if x> 0
1, if x= 0.
4x,if x< 0
For each of the following, find the limit. If the limit does not exist, so
state or use the symbol or – where appropriate. Hint: Sketch the graph of f.
(a) lim
x
0+
f(x)
(b) lim
x
0–
f(x)
(c) lim
x
0
f(x)
(d) lim
x
f(x)
(e) lim
x–
f(x)
79)
80)
The revenue function for a certain product is given by R(x) = 500x–6x2. Graph this
function in the window 0, 4 ×0, 2000 . Use TRACE to estimate lim
x
2.6
R(x).
80)
81)
The revenue R for a product is given by R(x) = 750x–x2 where x is the number of units
sold. Graph this function on your graphing calculator in several different viewing
rectangles. Discuss whether you think this function is continuous or not.
81)
82)
Find: lim e2
p
e
82)
83)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
1
x– 1
x2– 2x+ 1
83)
19
84)
The revenue R for a product is given by R(x) = 1750x–x2 where x is the number of units
sold. Is this function continuous for x= 100?
84)
85)
Find: lim
x
581
85)
86)
Find the value(s) of x for which f(x) =8x+ 1
x– 3 is discontinuous.
86)
87)
The demand function for a certain product is given by p(x) =50,000
(x+1)2 where p is the price
in dollars and x is the quantity sold. Find lim
x
p(x).
87)
88)
Use the definition of continuity to show that f(x) =4 –x is continuous at x= – 5.
88)
89)
The cost of purifying water is given by C=18,000
p– 2000 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Graph this function on
your graphing calculator and explore the graph as the percent of impurities gets very
small. Compare the cost of increasing the purity of the rate from 5% to 6% with the cost of
increasing the impurity from 50% to 51%. What does this mean?
89)
90)
Find the value(s) of x for which f(x) =x+ 2
x3– 4x is discontinuous.
90)
20