Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1)
lim
x
3x2– 4x+ 2
2x2+x+ 1
=
1)
A)
3
2
B)
2
C)
–
D)
–4
E)
2)
lim
x
0–
x2+ 4
x=
2)
A)
1
B)
C)
0
D)
4
E)
–
3)
lim
t
2
t2– t – 2
t2+3t – 10
=
3)
A)
–1
B)
C)
3
7
D)
1
5
E)
–
4)
Let f(x) =x(x+ 1)
x2– 1 . The only value(s) of x for which f is discontinuous is (are)
4)
A)
–1, 0, and 1.
B)
0.
C)
–1 and 1.
D)
–1.
E)
1.
5)
lim
x
3 =
5)
A)
3
B)
C)
–
D)
0
E)
does not exist
6)
lim
x–
4 –x2
1 –x=
6)
A)
0
B)
–
C)
–4
D)
E)
4
C
7)
The solution of 2x– 5
x+ 3 0 is
7)
A)
x–3, x5
2.
B)
–3 <x5
2.
C)
x< – 3, x5
2.
D)
–3 x5
2.
E)
x–3, x>5
2.
8)
lim
x–2(2 +3)=
8)
A)
–2 +3
B)
2 +3
C)
0
D)
3
E)
does not exist
9)
lim
q
2
4q2– q + 2
8=
9)
A)
1
4
B)
4
C)
5
D)
2
E)
3
C
10)
lim
x–3
x2+2x– 3
x2+ 7x + 12
=
10)
A)
–4
B)
–
C)
2
D)
0
E)
4
11)
The solution of x2+ 2x– 15
0 is
11)
A)
–5 x
3.
B)
x–5.
C)
x–5, x
3.
D)
x 3.
E)
x–5.
12)
lim
h
0
x2+ h
x + 2h=
12)
A)
B)
–
C)
0
D)
x
E)
does not exist
13)
lim
x
1–
7
1 – x =
13)
A)
B)
7
2
C)
0
D)
7
E)
–
A
14)
If f(x) =x + 2, if x> 0
x2+ 3, if x < 0 , then lim
x
0–
f(x) =
14)
A)
3
B)
C)
2
D)
0
E)
does not exist
15)
lim
x
4x2+2x+ 1
(x– 1)2=
15)
A)
4
B)
2
C)
1
D)
0
E)
16)
Let f(x) =4
x2+ 9 . The only value(s) of x for which f is discontinuous is (are)
16)
A)
–3 and 3.
B)
–3.
C)
4.
D)
3.
E)
none
17)
The solution of 9x2+ 6x+ 1 < 0 is
17)
A)
x< – 1
3.
B)
–1
3<x<1
3.
C)
x> – 1
3.
D)
all real numbers
E)
no solution
5
A
18)
lim
q
7q2+ 4q– 1
2q+ 3 =
18)
A)
7
2
B)
0
C)
–1
3
D)
–
E)
19)
If f(x) =x + 1, if x 1
x– 1, if x < 1 , then lim
x
1f(x) =
19)
A)
2
B)
0
C)
–
D)
E)
does not exist
20)
lim
x–5
x+ 3
5 –x=
20)
A)
–1
5
B)
0
C)
D)
–
E)
4
5
21)
The solution of x2– 5x– 14 < 0 is
21)
A)
x< – 2
B)
–2 <x< 7
C)
x> 7
D)
x< – 2, x> 7
E)
all real numbers
E
22)
The solution of (x+ 2)(x+ 4)(2x– 3) < 0 is
22)
A)
x< – 4, x>3
2.
B)
x< – 4, –2 <x<3
2.
C)
–2 <x<3
2.
D)
–4 <x< – 2, x>3
2.
E)
–4 <x< – 2.
23)
lim
x–
4 +x2
x2+x
=
23)
A)
0
B)
4
C)
1
D)
–
E)
24)
lim
x
3+
3x– 9 =
24)
A)
1
B)
3
C)
0
D)
9
E)
25)
The solution of x3–8x2> 0 is
25)
A)
x> 0.
B)
x> 8.
C)
x< 0, x> 8.
D)
0 <x< 8.
E)
x< 0.
26)
If f(x) = 3x– 7, then lim
h
0
f(x+h) –f(x)
h=
26)
A)
3x.
B)
0.
C)
3.
D)
–7.
E)
does not exist
27)
The solution of x2– 1
x2– 5x+ 6
> 0 is
27)
A)
x< – 1, 1 <x< 3.
B)
1 <x< 2, x> 3.
C)
x< 2, 2 <x3, x > 3.
D)
–1 <x< 1, 2 <x< 3.
E)
x< – 1, 1 <x< 2, x> 3.
28)
If f(x) =x,if x 2
2 –x,if x < 2 , then lim
x
2f(x) =
28)
A)
0
B)
1
C)
2
D)
E)
does not exist
C
29)
lim
x
1
2x2+ x – 3
x2+4x – 5
=
29)
A)
5
6
B)
2
5
C)
0
D)
3
5
E)
1
4
30)
The solution of (x+ 1)(x2– 1) < 0 is
30)
A)
x< – 1, –1 <x< 1.
B)
x> 1.
C)
x< 1.
D)
x< – 1, x> 1.
E)
–1 <x< 1.
31)
lim
x–
x– 1
4x2– 5
=
31)
A)
0
B)
C)
1
4
D)
1
5
E)
–
32)
If f(x) =3, if x> 4
1, if x 1 , then lim
x
4+
f(x) =
32)
A)
0
B)
1
C)
3
D)
4
E)
does not exist
A
33)
lim
x
5 –x2
(x4– 8x2+ 2)
=
33)
A)
5
2
B)
–
C)
0
D)
E)
5
34)
The solution of (x+ 9)(x– 1)(x– 5) 0 is
34)
A)
–9 x
1, x
5.
B)
x–9.
C)
x–9, 1 x
5.
D)
x 5.
E)
–9 x
1.
35)
Let f(x) =x2– 9
x2+ 2x+ 1 . The only value(s) of x for which f is discontinuous is (are)
35)
A)
1, –3 and 3.
B)
–1.
C)
–1 and 1.
D)
–1, 1, –3, and 3.
E)
–3 and 3.
36)
lim
x–1
x2+4x+ 3
x2– x – 2
=
36)
A)
0
B)
C)
–4
D)
–2
3
E)
1
37)
The solution of 2x2– 13x+ 6 0 is
37)
A)
x1
2, x 6.
B)
x 6.
C)
1
2x
6.
D)
x1
2.
E)
no solution
38)
lim
x
2+
4
x– 2 =
38)
A)
–
B)
C)
0
D)
4
E)
–2
39)
The solution of x+ 5
x2+ 2x– 8
0 is
39)
A)
x–5, –4 <x< 2.
B)
x–5, –4 x 2.
C)
x–4, x > 2.
D)
x= – 5, x> 2.
E)
–5 x< – 4, x> 2.
11
40)
If f(x) =2x,if x> 1
2, if x< 1 , then lim
x
1f(x) =
40)
A)
0
B)
1
C)
2
D)
E)
does not exist
41)
The solution of x2– 4
x< 0 is
41)
A)
–2 <x< 0, x> 2.
B)
–2 <x< 0, 0 <x< 2, x> 2.
C)
x< – 2, 0 <x< 2.
D)
–2 <x< 0, x> 0.
E)
–2 <x< 0, 0 <x< 2.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
42)
f(x) =
2 –x2if x> 1
–2 + 3xif 0 x
1
4 –x2if x< 0
Find
(a) lim
x
1+
f(x)
(b) lim
x
1–
f(x)
(c) lim
x
0–
f(x)
42)
C
43)
Find: lim
x
2f(x)
43)
44)
Find: lim
x
2
x2– 5x+ 2
3x– 4
44)
45)
The revenue R for a product is given by R(x) = 1750x–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.
45)
46)
Solve: x2– 6x+ 8
x2– 1
< 0
46)
13
47)
By looking at the graph, give the following limits.
(a) lim
x
1+
f(x)
(b) lim
x
1–
f(x)
(c) lim
x–1+
f(x)
47)
48)
The rate of change of productivity p (in number of units produced per hour) increases with
time on the job by the function p=50(t2+ 4t)
t2+ 3t+ 20. Find lim
t
8
p.
48)
49)
Find: lim
x–4
3 –x
x – 4
49)
50)
The cost C of producing x units of a certain product is given by
C(x) = 35,000 + 170x+0.8x2. Find lim
x
C(x) and discuss what this means.
50)
51)
Find the value(s) of x for which f(x) =x2,if x
2
3x,if x< 2 is discontinuous.
51)
52)
Find all points where y=f(x) =2 –x–x2
15 – 2x–x2 is discontinuous.
52)
53)
Let f(x) =
2 –x2if x> 1
–2 + 3xif 0 x
1
1 –x2if x< 0
Find all points of discontinuity for this function.
53)
54)
The demand function for a certain product is given by: p=15,000
(q+ 5)2, where p is the price
and q is the quantity demanded.
(a) Find the values of q for which p> 0.
(b) Find the values of q for which p< 0.
(c) What do the results of (a) and (b) mean and why does this make sense?
54)
55)
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
9
V(r).
55)
56)
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
6
V(r).
56)
57)
lim
x–
7
(1 + 3x)5
57)
58)
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. Graph V(r) in the standard viewing
rectangle, 0, 5 ×30, 50 and use TRACE to estimate lim
r
2.2 V(r).
58)
59)
Find: lim
x
6
x3–6x2
x– 6
59)
36
60)
A container manufacturer will make an open box by cutting a 3–inch square from each
corner of a square sheet of aluminum and then turning up the sides. The box is to contain
at least 300 in3. Find the dimensions of the smallest sheet of aluminum that can be used.
60)
61)
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.
61)
62)
Find: lim
x
7. If the limit does not exist, so state or use the symbol or – if appropriate.
62)
63)
Find lim
h
0
f(x+h) –f(x)
h where f(x) =2x+ 3
63)
64)
Use the definition of continuity to show f(x) = 2 is continuous at x= 5.
64)
45
65)
If the profit function for a certain business is given by: P(x) = 225x–3x2– 800, use the rule
about the limit of a polynomial function to determine lim
x
80
P(x).
65)
66)
The cost C of producing x units of a certain product is given by
C(x) = 50,000 + 200x+ 0.3x2. Find lim
x
C(x) and discuss what this means.
66)
67)
Consider the graph of f(x)
Determine the following limits:
(a) lim
x
2+f(x)(d) f(–2) (g) lim
x
1f(x)
(b) lim
x
2–
f(x)(e) lim
x
1+
f(x)(h) f(1)
(c) lim
x–2f(x)(f) lim
x
1–
f(x)
67)
68)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
2– 3x+5x3
7 – 8x+x2
68)
69)
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)
69)
70)
The demand function for a certain product is given by: p=10,000
(q+ 5)2, where p is the price
and q is the quantity demanded.
(a) Find the values of q for which p> 0.
(b) Find the values of q for which p< 0.
(c) What do the results of (a) and (b) mean and why does this make sense?
70)
71)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
1+
x– 1
x2– 2x+ 1
71)
72)
Find: lim
q–3
3q2–q– 5
10
72)
18
73)
Use the definition of continuity to show that f(x) =x2– 3x+ 1 is continuous at x= 2.
73)
74)
If f(x) = 2x+ 7, find lim
h
0
f(x+h) –f(x)
h by treating x as a constant.
74)
Answer:
2
Explanation:
75)
Find: lim
x–2+
f(x)
75)
Answer:
Explanation:
76)
Solve: x2–x– 6
0
76)
Answer:
Explanation:
77)
Solve: x+ 1
x– 3 < 0
77)
Answer:
Explanation:
78)
The rate of change of productivity p (in number of units produced per hour) increases with
time on the job by the function p=50(t2+ 4t)
t2+ 3t+ 20. Find lim
t
5
p.
78)
Answer:
Explanation:
79)
Find: lim
x
0
2
x. If the limit does not exist, so state or use the symbol or – if appropriate.
79)
Answer:
Explanation:
Answer:
Explanation:
80)
Find: lim
x
2x2– 4
6x3+2x2. If the limit does not exist, so state or use the symbol or – if
appropriate.
80)
81)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =4100(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
81)
82)
Find the value(s) of x for which f(x) =2x– 3
4x+ 8 is discontinuous.
82)
83)
Solve the inequality: x2– 1
x2+ 4
< 0.
83)
84)
The profit function for a certain business is given by: P(x) = 225x–3x2– 800. Graph this
function on your graphing calculator and use the evaluation function to determine
lim
x
43.9 P(x) using the rule about the limit of a polynomial function.
84)
85)
The cost C of producing x units of a certain product is given by C(x) = 10,000 + 50x+4x2.
Use your graphing calculator to explore lim
x
C(x) and discuss what this means.
85)
86)
Find: lim
x
0–
5
x. If the limit does not exist, so state or use the symbol or – if appropriate.
86)