87)
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?
87)
88)
Solve the inequality: 2 –x–x2
15 – 2x–x2
0.
88)
89)
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.
89)
90)
Find: lim
x–2
x2+ 4x+ 4
x2– 4
90)
91)
Find the value(s) of x for which f(x) =8x+ 1
x– 3 is discontinuous.
91)
92)
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).
92)
93)
The profit function for a product is given by P(x) = 28x+x2– 960. Determine for what
values of x the profit is positive.
93)
94)
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.
94)
95)
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.
95)
96)
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 determine lim
p
0C.
96)
97)
An open box is formed by cutting a square piece out of each corner of a 8–inch by 12–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)(12 – 2x).This problem only makes sense when this volume is positive.
Find the values of x for which the volume is positive.
97)
98)
Find: lim
t
2
t2+ 2t– 8
t– 2
98)
99)
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)
99)
100)
Find: lim
x
5
x2
h
100)
101)
Solve the inequality: (x– 2)ex
(x– 3) < 0.
101)
102)
Find: lim
x
2
x2+x– 6
x2+ 2x– 8
102)
103)
The cost of purifying water is given by C=10,000
p– 1250 where C is the cost in dollars and
p is the percent of impurities left in the water after purification. Find lim
p
0C.
103)
104)
The length of a material increases as it is heated up according to the equation l= 100 + 2x.
The rate at which the length is increasing is given by: lim
h
0
100 + 2(x+h) – (100 + 2x)
h.
Calculate this limit.
104)
105)
Find: Find lim
x
1
x2+ 4x– 5
x2+x– 2
105)
106)
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.
106)
107)
Find: lim e
x
5
107)
108)
Why can we say that f(x) =x5– 9x4 + 7x3– 5x2– 8x + 1 is continuous on ( , )?
108)
109)
Solve: x+ 4
x2– 4x+ 3
< 0
109)
110)
Solve: x2(x2– 9) < 0
110)
111)
Consider the graph of f(x)
Determine the following limits:
(a) lim
x
3+f(x)(d) f(3)
(b) lim
x
3–
f(x)(e) lim
x
f(x)
(c) lim
x
3f(x)(f) lim
x
f(x)
111)
112)
The demand function for a certain product is given by p(x) =50,000
(x+4)2 where p is the price
in dollars and x is the quantity sold. Find lim
x
p(x).
112)
113)
If f(x) = 6, find lim
h
0
f(x+h) –f(x)
h by treating x as a constant.
113)
114)
Use the definition of continuity to show that f(x) =5
x – 7 is continuous at x= 6.
114)
115)
By looking at the graph, give the following limits.
(a) lim
x–1+
f(x)
(b) lim
x–1–
f(x)
115)
116)
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)
116)
117)
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).
117)
118)
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?
118)
119)
Find: lim
x
4x2– 6x
x2+ 4x. If the limit does not exist, so state or use the symbol or – if
appropriate.
119)
120)
Find: lim
x
4
2x+ 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
120)
121)
Find: lim
x
4x4– 2x2– 3
x2– 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
121)
122)
If f(x) =4, if x>3
x+ 1, if x< 3 , find lim
x
3f(x). Hint: Sketch the graph of f.
122)
123)
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?
123)
124)
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?
124)
125)
Find: lim
x
581
125)
126)
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?
126)
127)
Find the value(s) of x for which f(x) =4x2+ 2x– 5 is discontinuous.
127)
128)
If f(x) =3 –x,if x> 2
3x– 5, if x< 2 , find lim
x
2f(x). Hint: Sketch the graph of f.
128)
129)
Find: lim (2x2– 4x+ 3)
x
3
129)
130)
The revenue function for a certain product is given by R(x) = 500x–6x2. Find lim
x
4R(x).
130)
131)
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.
131)
132)
Solve the inequality: x2+ 4
x2+ 1
> 0.
132)
133)
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).
133)
134)
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.
134)
135)
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.
135)
136)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
1– 2x+3x2
7 – 8x+ 11x3
136)
137)
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.
137)
138)
Find: lim
x–2
x+ 2
x2– 3
138)
139)
Find: lim
x
0–
x – 1
x. If the limit does not exist, so state or use the symbol or – if
appropriate.
139)
140)
Find: lim
x
2+
3
2 –x. If the limit does not exist, so state or use the symbol or – if
appropriate.
140)
141)
Find: lim
x–
x2+ 2x+ 3
5x5+ 4x. If the limit does not exist, so state or use the symbol or – if
appropriate.
141)
142)
The cost C of processing exhaust gases so that you remove all but x percent of the
pollutants is given by C(x) =7500(100 –x)
x. Find the values of x for which C< 0. Discuss
the meaning of the results.
142)
143)
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)
143)
Explanation:
144)
Solve the inequality: x2+ 1
x2– 4
< 0.
144)
Answer:
–2 <x< 2
Explanation:
145)
Find the value(s) of x for which f(x) =x– 2
x2+ 4x is discontinuous.
145)
Answer:
Explanation:
146)
Solve: x2– 4
x2< 0
146)
Answer:
–2<x<0, 0 <x< 2
Explanation:
147)
Use the definition of continuity to show that f(x) =4 –x is continuous at x= – 5.
147)
Answer:
Explanation:
148)
Find the value(s) of x for which f(x) =4x
3 is discontinuous.
148)
Answer:
Explanation:
Answer:
149)
Find: lim
x–1–
g(x)
149)
150)
Find: lim
x–
x3+ 1
x2+ 2 . If the limit does not exist, so state or use the symbol or – if
appropriate.
150)
151)
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)
151)
152)
An open box is formed by cutting a square piece out of each corner of a 12–inch by 10–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(12 – 2x)(10 – 2x). This problem only makes sense when this volume is positive.
Find the values of x for which the volume is positive.
152)
153)
Find the value(s) of x for which f(x) = 10 is discontinuous.
153)
154)
The profit function for a product is given by P(x) = 12x+x2– 4320. Determine for what
values of x the profit is positive.
154)
32
155)
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 . Find lim
x
y(x) and discuss what this means to the company.
155)
156)
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.
156)
157)
Find: lim
h
0
5 +h
x
157)
158)
Use the definition of continuity to show f(x) = 2x+ 3 is continuous at x= – 1.
158)
159)
Solve: x2– 3x+ 2
x2– 6x+ 9
> 0
159)
160)
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
20
P(x).
160)
161)
Find: lim
x
1+
4x– 1
3x. If the limit does not exist, so state or use the symbol or – if
appropriate.
161)
162)
Solve: (x+ 6)(x– 1)(x– 4)
0
162)
33
163)
Find: lim e2
p
e
163)
164)
Use the definition of continuity to show that f(x) =x+ 1
x– 1 is continuous at x= 3.
164)
165)
Find: Find lim
x
3
x2– 8x+ 15
3 + 2x–x2
165)
166)
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.
166)
167)
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).
167)
168)
Find: Find lim
x
2
x2– 4x+ 4
x2+x– 6
168)
169)
Find the following limit. If it is + or – or does not exist, then say so.
lim
x
2–
1– 2x
(x2+x– 6)
169)
34
170)
Find: lim
x–
2 – 9x
6. If the limit does not exist, so state or use the symbol or – if
appropriate.
170)
171)
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
4
p.
171)
172)
Find: lim
h
0
3(x+h) – 3x
h
172)
173)
If f(x) = 3x– 4, find lim
h
0
f(x+h) –f(x)
h by treating x as a constant.
173)
174)
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.
174)
175)
The revenue function for a certain product is given by R(x) = 500x–6x2. Find lim
x
0R(x).
175)
176)
Find: Find lim
x
1
x2– 2x+ 1
x2– 1
176)
177)
Solve the inequality: x3> 27.
177)
178)
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.
178)
179)
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
3.7
R(x).
179)
180)
Find: lim
p
4
p2– 7p+ 12
p2– 3p– 4
180)
181)
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).
181)
182)
Find lim
h
0
f(x+h) –f(x)
h where f(x) =1
2x+ 3
182)
183)
Find: lim
x–
2 –x
x2+ 3x. If the limit does not exist, so state or use the symbol or – if
appropriate.
183)
184)
Find the value(s) of x for which f(x) =1
1+ x2 is discontinuous.
184)
185)
Find: lim
x
1f(x)
185)
186)
Find: lim
x
4x3– 2
7 –2x3. If the limit does not exist, so state or use the symbol or – if
appropriate.
186)
187)
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?
187)
188)
Find: lim
x
3
5
x– 3 . If the limit does not exist, so state or use the symbol or – if
appropriate.
188)
189)
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).
189)