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Math 1431 – Homework 1
Section: _____________________
Instructions:
• Write your name and peoplesoft ID on each page before submitting. • Homework will NOT be accepted through email or in person. Homework must be submitted through
CourseWare BEFORE the deadline. • If the problem is from the text, the section number and problem number are in parentheses. • Use a blue or black pen or a pencil (dark). • Write your solutions in the spaces provided. You must show ALL work in order receive credit for a
problem. • Remember that your homework must be complete, neatly written and readable. • Submit this assignment at http://www.casa.uh.edu under "Assignments" and choose Homework 1.
1. (Section 1.2, Problem 15)
Name: ________________________________________
PSID: _________________________________________
2. (Section 1.3, Problem 8)
3. (Section 1.3, Problem 14)
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4. (Section 1.3, Problem 26)
5. (Section 1.3, Problem 28)
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6. (Section 1.3, Problem 40)
7. (Section 1.3, Problem 48)
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8. (Section 1.3, Problem 50)
9. (Section 1.3, Problem 54)
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10. (Section 1.3 Problem 68)
Suggested Homework: (Not to be turned in) Section 1.2: Problems 4, 14, 21, 22 Section 1.3: Problems 2,6, 15, 29, 36, 52, 55, 56, 57, 60, 69.
Math 1431 Page 1 of 5 Section 1.2 Exercises
Section 1.2 – Exercises
In Exercises 1-4, given the value of c and the graph of the function f , find lim ( ) x c
f x
.
1. c = 1 2. c = 2
3. c = 1 4. c = 4
Math 1431 Page 2 of 5 Section 1.2 Exercises
In Exercises 5-12, given the graph of a function f , use the graph to find (a) lim ( ) x c
f x
(b) lim ( ) x c
f x
(c) lim ( ) x c
f x
(d) ( )f c .
5. c = 1
6. c = 2
7. c = 2
8. c = 1
Math 1431 Page 3 of 5 Section 1.2 Exercises
9. c = 2
10. c = 3
11. c = 4
12. c = 1
Math 1431 Page 4 of 5 Section 1.2 Exercises
In Exercises 13-15, give the values of c for which lim ( )
x c f x
does not exist.
13.
14.
15.
2 , 2
( ) 8 2 , 2 4
4, 4
x x
f x x x
x
(Hint: draw the graph.)
Math 1431 Page 5 of 5 Section 1.2 Exercises
In Exercises 16-22, decide on intuitive grounds whether the indicated limit exists. If it does, find the limit.
16. 1
( ) 4 5, lim x
f x x f x
.
17. 2 0
( ) 1, lim x
f x x f x
.
18. 0
1 ( ) , lim
x f x f x
x .
19. 2
2
, 2 ( ) , lim
2 , 2 x x x
f x f x x x
.
20. 2
2
, 2 ( ) , lim
2 , 2 x x x
f x f x x x
.
21. 2
2
, 2 ( ) , lim
7 , 2 x x x
f x f x x x
.
22. 2
3
, 2 ( ) , lim
7 , 2 x x x
f x f x x x
.
Math 1431 Page 1 of 4 Section 1.3 Exercises
Section 1.3 – Exercises
In Exercises 1-32, evaluate the limit:
1. 2 2
lim 6 2 x
x x
2. 2 2
lim 6 2 x
x x
3. 4
5 lim
4x x
4. 23
3 lim
4 21x x
x x
5. lim x3
x 7 x 2 4x 21
6. 2
0
6 7 lim x
x x
x
7. 3
3 lim
18 6x x
x
8.
3
3
5 135 lim
3x x
x
9. 3
3
5 135 lim
3x x
x
10. 3 2
0
6 lim
2x x x
x
11. 2
20
6 lim
2x x x
x
12. 4
1
6 6 lim
2 2x x
x
13. 2
12 6 lim
2x x
x
14.
2
22
2 8 lim
4 4x x
x x
15. 5
| 5 | lim
5x x
x
16.
3
1 lim
3x x
17. 2
| | lim x
x
x
18. 8
8 lim x
x
x
19. 2 8
lim 64 x
x
20. 3
lim 6 x
21. 3
lim | 4 24 | x
x
22. 0
7 lim x
x x
23. 0
3 lim 6 x
x x
24. 2
0
3 lim 6 x
x x
25. 20
2 lim 6 x
x x
26.
4
2 lim
4x x
x
27. 4
4 lim
2x x
x
28. 2
0
4 1
lim 2
1 x
x
x
29. 2
0
2
4 1
lim 4
1 x
x
x
30. 2
5 3 lim
2 2 4x x
x x
Math 1431 Page 2 of 4 Section 1.3 Exercises
31. 2
2
5 5 lim
2 2 8x x
x x
32.
3
2
3 6 lim
2 3x x x
x
33. Evaluate 3
lim ( ) x
f x
, given that 6 , 3
( ) 18, 3
x x f x
x
.
34. Evaluate 0
lim ( ) x
f x
, given that 2 , 0
( ) 7, 0
x x f x
x x
.
35. Evaluate 3
lim ( ) x
f x
, given that 2
2 3, 3 ( )
, 3
x x f x
x x x
.
36. Evaluate 3
lim ( ) x
f x
, given that
2 , 3
( ) 8, 3
2 3, 3
x x
f x x
x x
.
37. Evaluate 1
lim ( ) x
f x
, given that 2 2, 1
( ) 1, 1
x x f x
x
38. Evaluate 2
lim ( ) x
f x
, given that 4 , 2
( ) 0, 2
x x f x
x
.
39. For 4
lim 3 12 x
x
, find the largest δ that works for ε = 0.1.
40. For 3
1 lim
6 2x x
, find the largest δ that works for ε = 0.01.
41. Given lim ( ) 6 x c
f x
, lim g( ) 2 x c
x
, and lim h( ) 0 x c
x
, evaluate the limit 1
lim ( ) ( )x c f x g x
.
42. Given lim ( ) 6 x c
f x
, lim g( ) 0 x c
x
, and lim h( ) 4 x c
x
, evaluate the limit 3lim ( ) x c
h x
.
43. Given that 2( ) 4f x x x . Evaluate the limit 1
( ) (1) lim
1x f x f
x
.
44. True or False: If lim ( ) ( ) x c
f x g x
exists but lim ( ) x c
f x
does not exist, then lim g( ) x c
x
does
not exist.
45. Given 1, is rational
( ) 1, is irrational.
x f x
x
, find
0 lim x
f x
.
46. Given 1, is integer
( ) 0, is not an integer.
x f x
x
, find 4
lim x
f x
.
47. Given ( ) x
f x x
, find 0
lim x
f x
.
Math 1431 Page 3 of 4 Section 1.3 Exercises
48. Given 9
( ) 9
x f x
x
, find
9 lim x
f x
.
49. Given 9
( ) 9
x f x
x
, find
10 lim x
f x
.
50. Given
2 , 2
( ) 3 , 2 5
2 1, 5
x x x
f x x x
x x
, find 2
lim x
f x
and 5
lim x
f x
.
In Exercises 51-62, find lim x
f x
and lim x
f x
. If these limits do not exist, state the
reason.
51. 5 2( ) 4 1f x x x .
52. 5 2
6
3 4 ( )
x x f x
x x
.
53. 5 4
4
2 3 ( )
x x f x
x x
.
54. 3 2
3
3 4 ( )
4 6
x x f x
x x
.
55. 2
2
5 4 ( )
2
x f x
x
.
56. 3
2 ( )
4 6
x x f x
x x
.
57. ( ) arctanf x x .
58. ( ) 5 xf x e .
59. ( ) cos(2 )f x x .
60. ( ) ln( )f x x .
61. ( ) sin
x f x
x .
62. ( ) 5sinf x x x .
Math 1431 Page 4 of 4 Section 1.3 Exercises
In Exercises 63-67, give an , proof for the following limits.
63. 2
lim 5 1 9 x
x
.
64. 3
lim 4 2 14 x
x
.
65. 2 4
lim 16 x
x
.
66. 3 1
lim 1 x
x
.
67. 2 1
lim 2 3 x
x x
.
In Exercises 68-71, find each limit. Notice that the limit is taken as 0h .
68. Find the limit: 2 2
0 lim h
x h x
h
.
69. Find the limit: 0
lim h
x h x
h
.
70. Find the limit: 0
1 1
lim h
x h x h
.
71. Find the limit: 3 3
0 lim h
x h x
h
.
72. a) Verify that if lim x c
f x L
, then lim x c
f x L
.
b) Give an example to show that the converse is false; find a function such that
lim x c
f x L
and lim x c
f x M L
.
c) Is it possible to find a function such that lim x c
f x
exists but lim x c
f x
does not exist?
