91)
By looking at the graph, give the following limits.
(a) lim
x–1+
f(x)
(b) lim
x–1–
f(x)
91)
92)
Find: lim
x
7. If the limit does not exist, so state or use the symbol or – if appropriate.
92)
93)
Solve: x2+ 2x– 3
x– 5
0
93)
94)
The demand function for a certain product is given by p(x) =10,000
(x+1)2 where p is the price
in dollars and x is the quantity sold. Find lim
x
p(x).
94)
95)
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)
95)
96)
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 5.90 , 5.95 , 5.99 , 6.01 ,
and 6.05 , determine lim
x
6x.
96)
97)
Solve: x2– 3x+ 2
x2– 6x+ 9
> 0
97)
98)
If f(x) = 3x– 4, find lim
h
0
f(x+h) –f(x)
h by treating x as a constant.
98)
99)
The revenue R for a product is given by R(x) = 8500x–x2 where x is the number of units
sold. Is this function continuous for x= 20?
99)
100)
Solve: x2– 4
x2< 0
100)
101)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =8300(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
101)
102)
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)
102)
103)
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.
103)
104)
The revenue R for a product is given by R(x) = 2500x–x2 where x is the number of units
sold. Is this function continuous for x= 20?
104)
105)
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.
105)
106)
Solve the inequality: x3> 27.
106)
107)
Find: Find lim
x
1
x2– 2x+ 1
x2– 1
107)
108)
The demand function for a certain product is given by p(x) =1000
(x+10)2 where p is the price
in dollars and x is the quantity sold. Graph this on your graphing calculator in the window
0,100 ×0,10 . Use the TRACE function to find lim
x
p(x). Describe in words what is
happening to the graph and what this means about the demand function.
108)
109)
Find the value(s) of x for which f(x) =4x2+ 2x– 5 is discontinuous.
109)
110)
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
120
P(x).
110)
111)
The cost of purifying water is given by C=40,000
p– 3750 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Find lim
p
0C.
111)
112)
If f(x) =3 –x,if x> 2
3x– 5, if x< 2 , find lim
x
2f(x). Hint: Sketch the graph of f.
112)
113)
Find the value(s) of x for which f(x) =x2– 1
x2– 2x– 8 is discontinuous.
113)
114)
Solve: x+ 4
x2– 4x+ 3
< 0
114)
115)
Find: lim
x
0–
5
x. If the limit does not exist, so state or use the symbol or – if appropriate.
115)
116)
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 3.90 , 3.95 , 3.99 , 4.01 ,
and 4.05 , determine lim
x
4x.
116)
117)
Solve: (x+ 6)(x– 1)(x– 4)
0
117)
118)
Use the definition of continuity to show f(x) = 2 is continuous at x= 5.
118)
119)
The demand function for a certain product is given by p(x) =8,000
(x+1)2 where p is the price
in dollars and x is the quantity sold. Find lim
x
p(x).
119)
120)
Suppose that the rent for an apartment in a city with rent control is $740 per month. Show
that the rent function R(t) = 740 is continuous at t= 3.
120)
121)
Solve the inequality: (x– 2)ex
(x– 3) < 0.
121)
122)
The demand function for a certain product is given by: p=20,000
(q+ 4)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?
122)
123)
Find: lim
x–1–
g(x)
123)
124)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
3 – 5e2–3x
11
124)
26
125)
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)
125)
126)
Find: lim
x
4f(x)
126)
127)
The revenue function for a certain product is given by R(x) = 500x–6x2. Find lim
x
7R(x).
127)
128)
Find: Find lim
x
2
x3– 8
x– 2
128)
129)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
2– 3x+5x3
7 – 8x+x2
129)
130)
Solve the inequality: 2 –x–x2
15 – 2x–x2
0.
130)
131)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =6200(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
131)
132)
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).
132)
133)
Find the value(s) of x for which f(x) =4x
3 is discontinuous.
133)
134)
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.
134)
135)
Find the value(s) of x for which f(x) =2x– 3
4x+ 8 is discontinuous.
135)
136)
Find: lim
x–
2 – 9x
6. If the limit does not exist, so state or use the symbol or – if
appropriate.
136)
137)
A math tutor charges $30 for the first hour of work at your house and $20 for every hour
(or fraction thereof) afterwards. The function for what the session will cost you is given by:
f(x) =
$30 0 <x
1
$50 1 <x
2
$70 2 <x
3
$90 3 <x
4
(a) Find lim
x
1f(x)
(b) Find lim
x
2.5 f(x)
(c) Find lim
x
f(x)
137)
138)
The demand function for a certain product is given by: p=5000
(q+ 3)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?
138)
139)
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.
139)
140)
Find: Find lim
x
1
x2+ 4x– 5
x2+x– 2
140)
141)
Let f(x) =5, if x> 4
x,if x 4 . 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
4+
f(x)
(b) lim
x
4–
f(x)
(c) lim
x
4
f(x)
(d) lim
x
f(x)
(e) lim
x–
f(x)
141)
142)
Find: lim
x
3f(x)
142)
143)
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).
143)
144)
lim
x–
7
(1 + 3x)5
144)
145)
An open box is formed by cutting a square piece out of each corner of a 8–inch by 8–inch
piece of metal. If each side of the squares cut out is x inches long, the volume of the box is
given by
V(x) =x(8 – 2x)(8 – 2x). This problem only makes sense when this volume is positive. Find
the values of x for which the volume is positive.
145)
146)
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) =800x
x+ 10 . Find lim
x
y(x) and discuss what this means to the company.
146)
147)
If f(x) = 6, find lim
h
0
f(x+h) –f(x)
h by treating x as a constant.
147)
148)
Solve: x2– 2x– 3 < 0
148)
149)
The profit function for a product is given by P(x) = 35x+x2– 4950. Determine for what
values of x the profit is positive.
149)
31
150)
A lawyer charges $300 for the first hour of work at your house and $100 for every hour (or
fraction thereof) afterwards. The function for what the visit will cost you is given by:
f(x) =
$300 0 <x
1
$400 1 <x
2
$500 2 <x
3
$600 3 <x
4
(a) Find lim
x
1f(x)
(b) Find lim
x
2.5 f(x)
(c) Find lim
x
f(x)
150)
151)
Find: lim
x–1+
1
x+ 1 . If the limit does not exist, so state or use the symbol or – if
appropriate.
151)
152)
The cost C of producing x units of a certain product is given by C(x) = 3,000 + 80x+0.1x2.
Find lim
x
C(x) and discuss what this means.
152)
153)
Solve the inequality: x2+ 1
x2– 4
< 0.
153)
154)
Find: lim
x–
x2+ 2x+ 3
5x5+ 4x. If the limit does not exist, so state or use the symbol or – if
appropriate.
154)
155)
Solve: x2
x2+ 1
0
155)
156)
Find the value(s) of x for which f(x) =2x
x2– 3x + 2 is discontinuous.
156)
157)
Use the definition of continuity to show that f(x) =x2– 3x+ 1 is continuous at x= 2.
157)
158)
Let f(x) =2x,if x 1
x,if x< 1 . 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
1–
f(x)
(b) lim
x
1+
f(x)
(c) lim
x
1
f(x)
(d) lim
x
f(x)
(e) lim
x–
f(x)
158)
159)
Find: lim e
x
5
159)
160)
Solve the inequality: x2+x
2.
160)
33
161)
Find: lim
x
1f(x)
161)
162)
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?
162)
163)
Find: lim
x
3+
x2x2– 9. If the limit does not exist, so state or use the symbol or – if
appropriate.
163)
164)
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).
164)
165)
Find: lim
x–2+
f(x)
165)
166)
Solve: x+ 1
x– 3 < 0
166)
167)
Let f(x) =
1
3+xif x 0
2
5+xif x< 0
Find all points where this function f(x) is not continuous.
167)
168)
Find: lim
x
2
x2+x– 6
x2+ 2x– 8
168)
169)
The profit function for a product is given by P(x) = 28x+x2– 960. Determine for what
values of x the profit is positive.
169)
170)
Find the value(s) of x for which f(x) = 10 is discontinuous.
170)
171)
Suppose that the rent for an apartment in a city with rent control is $480 per month. Show
that the rent function R(t) = 480 is continuous at t= 7.
171)
172)
Find: lim
x
2f(x)
172)
173)
Consumers will purchase q units of a certain product when the price is 340 – 2q dollars per
unit. How many units must be sold in order that sales revenue be no less than $14,000?
173)
174)
Solve the inequality: xex2–5> 0.
174)
175)
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?
175)
176)
Find: Find lim
x
3
x2– 8x+ 15
3 + 2x–x2
176)
36
177)
Find: Find lim
x
2
x2– 4x+ 4
x2+x– 6
177)
178)
Use your calculator to complete the table, and use your results to estimate the given limit.
lim
x
3
x2– 9
x– 3
x3.1 3.01 3.001 2.999 2.99 2.9
f(x)
178)
179)
Find: lim
x
3
5
x– 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
179)
180)
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
50
P(x).
180)
181)
Solve the inequality: x2– 1
x2+ 4
< 0.
181)
182)
Let f(x) =
2 –x2if x> 1
–2 + 3xif 0 x
1
1 –x2if x< 0
Find all points of discontinuity for this function.
182)