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Open the file attached and select the exercise you wish to submit. Complete a handwritten detailed worked-out solution, scan or photograph your solution and post for grading. CHOSE ONLY ONE MATH PROBLEM TO WORK ON, THIS PROBLEM MUST BE DONE ON A SHEET OF PAPER TO SHOW ALL STEPS OF HOW YOU CAME TO YOUR ANSWER
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Assignment # 1 Sec 1.1
1. Consider the function f and its graph given to the right.
Find a) lim f(x); b) lim f(x); c) lim f(x).
x →4 −
f(x) =
x − 2, for x ≤ 1
x + 1, for x > 1
x →4
x →4 +
If a limit does not exist, state that fact.
8
a) Select the correct choice below and, if necessary, fill in
the answer box to complete your choice.
y
6
4
A.
2
lim f(x) =
x
x →4 −
-8
B. The limit does not exist.
-6
-4
-2
b) Select the correct choice below and, if necessary, fill in
the answer box to complete your choice.
A.
2
4
6
8
-2
-4
-6
-8
lim f(x) =
x →4 +
B. The limit does not exist.
c) Select the correct choice below and, if necessary, fill in
the answer box to complete your choice.
A.
lim f(x) =
x →4
B. The limit does not exist.
2. Consider the function f and its graph given to the right.
Find a)
lim
g(x); b)
x→ − 1 −
lim
g(x); c) lim g(x).
x→ − 1 +
x + 6,
g(x) =
for x < − 1 − x + 2, for x ≥ − 1 x→ − 1 If a limit does not exist, state that fact. 8 a) Select the correct choice below and fill in any answer boxes in your choice. 6 A. g(x) = (Type an integer or a simplified fraction.) B. The limit does not exist. b) Select the correct choice below and fill in any answer boxes in your choice. lim g(x) = x→ − 1 + (Type an integer or a simplified fraction.) B. The limit does not exist. c) Select the correct choice below and fill in any answer boxes in your choice. A. 4 2 lim x→ − 1 − A. y lim g(x) = x→ − 1 (Type an integer or a simplified fraction.) B. The limit does not exist. x -8 -6 -4 -2 2 -2 -4 -6 -8 4 6 8 3. Decide from the graph whether a limit exists. If a limit exists, find its value. 10 y 8 lim F(x) 6 x→ − 2 4 What is the limit? Select the correct choice below and fill in any answer boxes in your choice. 2 -10 -8 -6 -4 -2 A. The limit is . x 2 -2 4 6 8 10 -4 B. The limit does not exist. -6 y = F(x) -8 -10 4. Use the graph of G shown to the right to find the limit. When necessary, state that the limit does not exist. 8 6 lim G(x) x →3 4 2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x -8 A. lim G(x) = -6 -4 -2 2 4 6 8 -2 (Type an x →3 -4 integer or a simplified fraction.) -6 B. The limit does not exist. -8 5. Use the graph of G shown to the right to find the limit. If necessary, state that the limit does not exist. 8 y 6 lim G(x) x →3 + 4 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. y 2 x -8 -6 -4 -2 2 -2 lim G(x) = x →3 + B. The limit does not exist. -4 -6 -8 4 6 8 6. Use the graph of f, shown on the right, to find the given limit. When necessary, state that the limit does not exist. 8 y 4 lim f(x) x x →4 -10 -5 5 10 -4 -8 Select the correct choice below and fill in any answer boxes in your choice. A. lim f(x) = (Type an integer or a simplified fraction.) x →4 B. The limit does not exist. 7. Use a table and/or graph to decide whether the limit exists. If a limit exists, find its value. lim 2x − 9 x →5 What is the limit? Select the correct choice below and fill in any answer boxes in your choice. A. The limit is . (Simplify your answer.) B. The limit does not exist. 8. Graph the given function and then find the specified limits. If necessary, state that the limit does not exist. 2 g(x) = x − 5; lim g(x) and lim g(x) x →3 A. x→ − 2 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. Choose the correct graph below. 10 -10 lim g(x) = B. The limit does not exist. 10 10 -10 10 -10 C. D. 10 y 10 x -10 10 (Simplify your answer.) B. The limit does not exist. -10 y x -10 10 lim g(x) = x→ − 2 y x -10 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. y x x →3 (Simplify your answer.) B. -10 9. Graph the following function and then find the specified limits. When necessary, state that the limit does not exist. G(x) = 5 ; find lim G(x) and lim G(x) x+4 x→ − 3 x→ − 4 Choose the correct graph below. A. B. 10 y C. 10 x -10 10 y D. 10 x -10 -10 10 -10 y 10 x -10 10 -10 y x -10 10 -10 Find lim G(x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x→ − 3 A. lim G(x) = (Type an integer or a simplified fraction.) x→ − 3 B. The limit does not exist. Find lim G(x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. x→ − 4 A. lim G(x) = x→ − 4 B. The limit does not exist. (Type an integer or a simplified fraction.) 10. Graph the following function and then find the specified limits. When necessary, state that the limit does not exist. f(x) = 1 + 1; find lim f(x) and lim f(x) x x →∞ x →0 Choose the correct graph below. A. B. 10 y C. 10 y 10 x -8 8 D. y 10 x -8 -10 8 -10 y x -8 8 -10 x -8 8 -10 Find lim f(x). Select the correct choice below and fill in any answer boxes in your choice. x →∞ A. lim f(x) = (Type an integer or a simplified fraction.) x →∞ B. The limit does not exist. Find lim f(x). Select the correct choice below and fill in any answer boxes in your choice. x →0 A. lim f(x) = x →0 B. The limit does not exist. (Type an integer or a simplified fraction.) 11. Find lim f(x), lim f(x), and lim f(x) for the given function. x →6 − f(x) = x →6 x →6 + x+5 for x ≤ 6 5x − 21 for x > 6
Select the correct choice below and fill in any answer boxes in your choice.
A.
lim f(x) =
(Simplify your answer.)
x →6 −
B.
lim f(x) does not exist.
x →6 −
Select the correct choice below and fill in any answer boxes in your choice.
A.
lim f(x) =
(Simplify your answer.)
x →6 +
B.
lim f(x) does not exist.
x →6 +
What is the value of lim f(x)?
x →6
A. 2
B. 11
C. 9
D. The limit does not exist.
12. Determine lim f(x), lim f(x), and lim f(x), if it exists.
x →c +
2−x
c = 3, f(x) =
x →c
x →c −
x<3 x +1 x>3
3
lim f(x) =
x →c +
lim f(x) =
x →c −
What is lim f(x)?
x →c
A. The limit is c.
B. The limit is − 1.
C. The limit does not exist.
D. The limit is 2.
13. Graph the following function and then find the specified limit. When necessary, state that the limit does not exist.
2
for x < 1, x , G(x) = Find lim G(x). x →1 − x + 2, for x > 1.
Choose the correct graph below.
A.
B.
10
C.
y
10
y
10
x
-5
D.
y
10
x
5
-5
x
5
-10
y
-5
x
5
-10
-5
5
-10
-10
Find lim G(x). Select the correct choice below and fill in any answer boxes in your choice.
x →1
A.
lim G(x) =
(Type an integer or a simplified fraction.)
x →1
B. The limit does not exist.
14. Graph the following function and then find the specified limits. When necessary, state that the limit does not exist.
x − 1 if x < 2 f(x) = 1 if 2 ≤ x ≤ 6 ; find lim f(x) and lim f(x) x + 1 if x > 6
x →2
x →6
Choose the correct graph below.
A.
15
B.
y
15
x
-5
15
-5
C.
y
15
x
-5
-5
15
D.
y
15
y
x
-5
-5
15
x
-5
-5
Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A.
lim =
x →2
B. The limit is not − ∞ or ∞ and does not exist.
Select the correct choice and, if necessary, fill in the answer box to complete your choice.
A.
lim =
x →6
B. The limit is not − ∞ or ∞ and does not exist.
15
Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
A.
lim
C(x) = $
x→0.94 −
(Simplify your answer. Do not include
the $ symbol in your answer.)
B. The limit does not exist.
Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
A.
lim
C(x) = $
x→0.94 +
(Simplify your answer. Do not include
the $ symbol in your answer.)
B. The limit does not exist.
Select the correct choice below and, if necessary, fill in the
answer box to complete your choice.
A.
lim C(x) = $
x→0.94
(Simplify your answer. Do not include
the $ symbol in your answer.)
B. The limit does not exist.
5.2
4.8
Cost of taxi (in dollars)
15. In a particular city, taxicabs charge passengers $2.80 for
entering a cab and then $0.40 for each one-fifth of a mile (or
fraction thereof) traveled. If x represents the distance
traveled in miles, then C(x) is the cost of the taxi fare, where
C(x) = $2.80 if x = 0, C(x) = $3.20 if 0 < x ≤ 0.2, and so on. The graph of C is shown to the right. Use the graph to find each of the following limits, if it exists. 4.4 4 3.6 3.2 2.8 0 0.2 0.4 0.6 0.8 Distance traveled (in miles) 1 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim p(x) = $ x →1 − (Simplify your answer. Do not include the $ symbol in your answer.) B. The limit does not exist. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim p(x) = $ x →1 + (Simplify your answer. Do not include the $ symbol in your answer.) B. The limit does not exist. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. lim p(x) = $ x →1 (Simplify your answer. Do not include the $ symbol in your answer.) B. The limit does not exist. 1.51 Postage cost (in dollars) 16. The cost of sending a large envelope via first-class mail is $0.95 for the first ounce and $0.14 for each additional ounce (or fraction thereof). If x represents the weight of a large envelope, in ounces, then p(x) is the cost of mailing it, where p(x) = $0.95 if 0 < x ≤ 1, p(x) = $1.09 if 1 < x ≤ 2, and so on, up through 13 ounces. The graph of p is shown to the right. Use the graph to find each of the following limits, if it exists. 1.37 1.23 1.09 0.95 0 1 2 3 4 Weight of envelope (in ounces) 5 17. The federal tax rate for heads of household is given in the graph shown to the right. Use the graph to find lim r(x), lim r(x), and lim r(x). x→8675 + Find r(x). Select the correct choice below and, if x→8675 − necessary, fill in the answer box to complete your choice. A. lim r(x) = % Tax rate (percentage) x→8675 x→8675 − lim 36.00 24.00 12.00 x→8675 − (Simplify your answer. Do not include the % symbol in your answer.) B. The limit does not exist. Find lim r(x). Select the correct choice below and, if x→8675 + necessary, fill in the answer box to complete your choice. A. lim r(x) = % x→8675 + (Simplify your answer. Do not include the % symbol in your answer.) B. The limit does not exist. Find lim r(x). Select the correct choice below and, if x→8675 necessary, fill in the answer box to complete your choice. A. lim r(x) = % x→8675 (Simplify your answer. Do not include the % symbol in your answer.) B. The limit does not exist. 0.00 8675 39337 89816 Taxable income (in dollars) 18. The federal tax rate for heads of household is given in the graph shown to the right. Use the graph to find lim r(x) and lim r(x). Find x→10,193 lim r(x).Select the correct choice below and, if x→53,000 necessary, fill in the answer box to complete your choice. A. lim r(x) = % Tax rate (percentage) x→53,000 18 12 6 x→53,000 (Simplify your answer. Do not include the % symbol in your answer.) B. The limit does not exist. Find lim r(x). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. lim r(x) = % x→10,193 (Simplify your answer. Do not include the % symbol in your answer.) B. The limit does not exist. 19. Determine the value for c so that lim f(x) exists. x →2 f(x) = 1 x + c, for x < 2 2 − x + 8, for x > 2
The value of c is
.
20. Determine the value for c so that lim f(x) exists.
x →4
f(x) =
2
x − 10, for x < 4 2 − x + c, for x > 4
Tha value of c is
10,193
43,731
Taxable income (in dollars)
x→10,193
A.
0
.
Assignment # 2 Sec 1.2
1. Find the given limit.
2
x −2
lim
3−x
x→ − 3
2
x −2
=
3−x
x→ − 3
(Simplify your answer.)
lim
2. Find the given limit.
2
x − 16
x+4
x→ − 4
lim
Select the correct choice below and fill in the answer box within your choice.
A.
2
x − 16
=
x→ − 4 x + 4
lim
(Simplify your answer.)
B. The limit does not exist.
3. Use algebra and the properties of limits as needed to find the given limit. If the limit does not exist, say so.
2
x + 3x − 270
lim
2
x − 225
x→15
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
2
A.
x + 3x − 270
lim
2
x − 225
x→15
=
(Type an integer or a simplified fraction.)
B. The limit does not exist.
4. Use algebra and the properties of limits as needed to find the given limit. If the limit does not exist, say so.
2
x + 3x − 340
lim
x→17
2
x − 289
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
2
A.
lim
x→17
x + 3x − 340
2
x − 289
=
B. The limit does not exist.
(Type an integer or a simplified fraction.)
5. Use algebra and the properties of limits as needed to find the given limit. If the limit does not exist, say so.
x − 13
lim
x→169
x − 169
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
x − 13
lim
x − 169
x→169
=
(Type an integer or a simplified fraction.)
B. The limit does not exist.
6. Simplify the function algebraically and find the limit.
2
x + 4x − 5
lim
2
x→1 x − 2x + 1
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
2
A.
x + 4x − 5
lim 2
=
x→1 x − 2x + 1
(Simplify your answer.)
B. The limit does not exist.
7. Find the indicated limit.
lim
9x − 8
x →1
Select the correct choice below and fill in the answer box within your choice.
A.
lim
9x − 8 =
x →1
B. The limit does not exist.
(Type an exact answer, using radicals as needed.)
8.
Use the graph to find the following limits
and function value.
y
4
a. lim f(x)
x→2

2
x
b. lim f(x)
x→2
+
-2
c. lim f(x)
2
4
-2
x→2
d. f(2)
a. Find the limit. Select the correct choice below and fill in any answer boxes in your choice.
A.
lim f(x) =
x→2
(Type an integer.)

B. The limit does not exist.
b. Find the limit. Select the correct choice below and fill in any answer boxes in your choice.
A.
lim f(x) =
x→2
(Type an integer.)
+
B. The limit does not exist.
c. Find the limit. Select the correct choice below and fill in any answer boxes in your choice.
A. lim f(x) =
(Type an integer.)
x→2
B. The limit does not exist.
d. Find the function value. Select the correct choice below and fill in any answer boxes in your choice.
A. f(2) =
(Type an integer.)
B. The answer is undefined.
9. Use the graph and the function to find the following.
a) Find lim k(x).
x →1
b) Find k(1).
c) Is k continuous at x = 1?
y
5
4
3
2
1
-5-4 -3 -2 -1
-1
-2
-3
-4
-5
a) Select the correct choice below and fill in any answer boxes in your choice.
A.
lim k(x) =
x →1
(Round to the nearest integer as needed.)
B. The limit does not exist.
b) Select the correct choice below and fill in any answer boxes in your choice.
A. k(1) =
(Round to the nearest integer as needed.)
B. The function is undefined at x = 1.
c) Is k continuous at x = 1?
Yes
No
y = k(x)
x
1 2 3 4 5
10. Use the graph to answer these questions.
y
10
a. Find lim f(x) and lim f(x).
x →4 +
x →4 −
b. Find lim f(x).
y = f(x)
x →4
c. Find f(4).
d. Is f(x) continuous at x = 4? Why or why not?
0
a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
lim f(x) =
(Simplify your answer.)
x →4 +
B. The limit does not exist.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
lim f(x) =
(Simplify your answer.)
x →4 −
B. The limit does not exist.
b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
lim f(x) =
(Simplify your answer.)
x →4
B. The limit does not exist.
c. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. f(4) =
(Simplify your answer.)
B. The function is undefined at x = 4.
d. Is f(x) continuous at x = 4? Why or why not?
A. No, f(x) is not continuous at x = 4 because lim f(x) does not exist.
x →4
B. Yes, f(x) is continuous at x = 4 because f(4) exists.
C. No, f(x) is not continuous at x = 4 because f(4) is undefined.
D. Yes, f(x) is continuous at x = 4 because lim f(x) = f(4).
x →4
x
0
10
11.
Is the function given by f(x) =
1
x + 3, for x ≤ 3,
3
continuous at x = 3? Why or why not?
4x − 8, for x > 3,
Choose the correct answer below.
A. The given function is continuous at x = 3 because f(3) does not exist.
B. The given function is not continuous at x = 3 because the limit does not exist.
C. The given function is not continuous at x = 3 because f(3) = lim f(x) = 4.
x →3
D. The given function is continuous at x = 3 because f(3) = lim f(x) = 4.
x →3
12. Is the function g(x) continuous at x = 2?
g(x) =
1
x + 1, for x ≤ 2
2
4x − 6,
for x > 2
Choose the correct answer below.
Yes
No
13. Is the function given by
g(x) =
1
x + 4,
3
for x < 6, − x + 12, for x > 6,
continuous at x = 6? Why or why not?
Select all that apply.
A. The function is continuous at x = 6 because lim g(x) is equal to g(6).
x →6
B. The function is continuous at x = 6 because g(6) exists.
C. The function is continuous at x = 6 because lim g(x) exists.
x →6
D. The function is not continuous at x = 6 because g(6) does not exist.
E. The function is not continuous at x = 6 because lim g(x) does not exist.
x →6
14. Is the function g(x) continuous at x = 4?
2
G(x) =
x − 16
, for x ≠ 4
x−4
6,
for x = 4
Choose the correct answer below.
Yes
No
15.
2
Is the function given by f(x) =
2x − 5x − 25
if x < 5, continuous at x = 5? Why or why not? x−5 6x − 15 if x ≥ 5, Choose the correct answer below. A. The given function is continuous at x = 5 because lim f(x) = f(5) = 15. x →5 B. The given function is continuous at x = 5 because f(5) does not exist. C. The given function is not continuous at x = 5 because the limit does not exist. D. The given function is not continuous at x = 5 because lim f(x) = f(5) = 15. x →5 16. 1 Is the function given by g(x) = 2 continuous at x = 3? Why or why not? x − 9x + 18 Select all that apply. A. The function is continuous at x = 3 because lim g(x) is equal to g(3). x →3 B. The function is not continuous at x = 3 because lim g(x) does not exist. x →3 C. The function is not continuous at x = 3 because g(3) does not exist. D. The function is continuous at x = 3 because lim g(x) exists. x →3 E. The function is continuous at x = 3 because g(3) exists. 17. Is the function given by G(x) = 1 continuous over the interval (1,∞)? Why or why not? x−7 Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. No, the function is not continuous at x = (Use a comma to separate answers as needed.) . B. Yes, the function is continuous over (1,∞) because lim G(x) = G(a) for all a in (1,∞). x →a 18. The Candy Factory sells candy by the pound, charging $1.60 per pound for quantities up to and including 20 pounds. Above 20 pounds, the Candy Factory charges $1.40 per pound for the entire quantity. If x represents the number of pounds, the price function is as follows. 1.60x for x ≤ 20 p(x) = Find 1.40x for x > 20
lim p(x),
x→20 −
Find
lim p(x), and lim p(x).
x→20
x→20 +
lim p(x).
x→20 −
lim p(x) =
(Type an integer or a decimal.)
x→20 −
Find
lim p(x).

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