AP Calculus AB
Unit 1 Homework Packet – Limits & Continuity
Name:________________________________
Per:__________
Below are tables of values for given types of functions. For each table, the type of function represented by the
table is given. Use your knowledge of the numerical behavior of each type of function to find the indicated
limits. For limits that do not exist, write D.N.E.
1. Exponential Function
a) lim H ( x) =
x 
b) lim H ( x) =
x 1
c) lim H ( x) =
x 
2. Rational Function
lim G ( x ) =
a) lim G( x) =
b)
d) lim G( x) =
e) lim G ( x ) =
g) lim G( x) =
h) lim G( x) =
x 
x 2
x1
x  2 
x 1
c)
lim G ( x ) =
x  2 
f) lim G ( x ) =
x 1
x 
3. Rational Function
a) lim H ( x) =
b) lim H ( x) =
d) lim G ( x) =
e) lim G( x) =
x 
x4
x1
x4
c) lim H ( x) =
x4
If they exist, determine the indicated values below each graph. For limits that do not exist, write D.N.E.
4. The graph of h(x) is given.
5. The graph of g(x) is given.
a) lim h( x ) =
b) lim h( x ) =
a) lim g ( x ) =
b) lim g ( x) =
c) lim h( x) =
d) h(1) =
c) lim g ( x) =
d) lim g ( x) =
e) h(–2) =
f) lim h( x) =
e) g(4) =
f) lim g ( x) =
x 1
x 1
x1
x 2
6. The graph of f(x) is given.
x 0
x 
x 1
x 4
x3
7. The graph of q(x) is given.
a) lim f ( x) =
b) lim f ( x) =
a) lim q( x) =
b) lim q( x) =
c) lim f ( x) =
d) lim f ( x ) =
c) lim q( x) =
d) lim q( x) =
e) lim f ( x) =
f) lim f ( x) 
e) q(–3) =
f) q(4) =
x0
x 
x 1
x 
x 1
x 1
x0
x4
x 3
x 4
Given the graph of the function, g(x), below, determine if the statements are true or false. For statements that
are false, explain why.
8. lim g ( x)  2
x 1
9. lim g ( x) exists for every value of c on the interval (–1, 1).
x c
10. lim g ( x) does not exist.
x 2
Sketch a graph of a function that fits the requirements described below.
11.
lim f ( x)  3
x 1
lim f ( x)  1
x 1
f(1) = 1
12.
lim
x  2 
f ( x)  
lim
x  2 
f ( x)  
f(2) is undefined but lim f ( x) exists.
x 2
13. In exercise 11, does lim f ( x) exist? Explain why or why not.
x1
Find the value of each limit. For a limit that does not exist, state why.
14.
16.
lim 3 x 2 (2 x  1)
x   12
lim ( x  6)
2
x  2
3
15.
lim x 3  2 x 2  3x  3
x  1
2
17. lim x  5 x  6
x2
x2
( x  4) 2 16
x
x 0
18. lim  2 tan 
19. lim
20. lim x21
2
21. lim x 23 x  2
  6
x 1 x 1
x2
x 4
3
2
22. lim 5 x4  8 x 2
x  0 3 x 16 x
23. lim
(2  x) 3  8
x
x 0
25. lim
24. lim
26.
lim
h®0
f (x+h)- f (x)
if f(x) = 3x2 – 2x
h
x 0
1 1
x2 2
x
( x  h) 2  2( x  h)  3  ( x 2  2 x  3)
h
h 0
2 x 2  4 x , x  2

27. lim f ( x ) if f ( x)  
x
x2
4 sin 4 , x  2
 
3 
28. lim e x cos x
x 3
30.
lim x  3
x 3 
32. lim
x 0
x 3
1 1
x 2 2
x
x 3 2
x 1 x 1
29. lim
31.
2x 2 9x 9
x 2 9
x  3 
33.
x  2
2  x,
lim  2
x  2
 x  2 x, x  2
lim
34. If lim f ( x)  2 and lim g ( x)  4 , find each of the following limits. Show your analysis applying
x 3
x 3
the properties of limits.
5 f ( x) 
a. lim 
x  3 g ( x ) 
b. lim  f ( x)  2 g ( x)
x 3
4 f ( x)
x 3
e. lim 3 f ( x)  g ( x)
g ( x)
x 3 8
d. lim
c. lim
x 3
f ( x) g ( x) 
f. lim 
12


x  3
35. If lim f ( x)  0 and lim g ( x)  3 , find each of the following limits. Show your analysis applying
x4
x4
the properties of limits.
g ( x) 
a. lim 
f

x  4  ( x ) 1 
d. lim g ( x)  3
x4
b. lim xf ( x)
x 4
e. lim g 2 ( x)
x 4
Find the limit of each of the following exponential functions. Sketch a graph of each function to aid in your
determination of the limit. CALCULATOR PERMITTED
36. lim  (0.5)  x  2  3
x 
37. lim 2  x  2  3
38.
x 
39. Using the graph of g(x) pictured to the right, find each of the following
limits.
a. lim g ( x)  _________
b.
lim g ( x)  _________
d.
x 
c.
x  1
lim g ( x)  _________
x  
lim g ( x)  _________
x  3
Find the value of each limit. For a limit that does not exist, state why.
2
40. lim cos 
  2 1 sin 
41. lim x  sin x
x 0
x
4 
x 2
lim  1
3
x  
2 x 2  3x,
x3

42. lim 
x 38  cos x , x  3
3

 
44.
lim sin2 x
x 0 2x  x
46. lim sin 2 x
x0 6x
43. lim 2 sin 3
 0

45. lim 5 x  sin 3 x
x 0
x
47. lim 2 sin 4 x
x 0
3x
48.
lim cos  tan 
 0
3
50. lim cos 
   cot 
2
3
52. lim c  27
c 3 c  3
49. lim 3  3 cos 
 0

1 tan 
  0 sin   cos 
51. lim
x -2
53. lim
x®4 x - 4
Use the graph of F(x) to the right to find each of the
indicated values below. If a value does not exist, state
why.
54. lim F ( x)  _________
x 
55. lim F ( x)  _________
x 2
56.
57.
58.
lim F ( x)  _________
x 2
lim F ( x)  _________
x 2
lim F ( x)  _________
x 2
lim F ( x)  _________
61. F(–5) = _________
lim F ( x)  _________
64.
59. F(–4.5) =_________
60.
62. F(1) = _________
63.
65. lim F ( x)  _________
66.
68. F(–2) = _________
69. F(–3) = _________
x 1
x  5
x  1
lim
x   3
F ( x)  _________ 67.
lim F ( x)  ______
x  1
lim
x   3
F ( x)  _____
Determine if the function is continuous at each of the
indicated values below. Use the three part definition
of continuity to perform your analysis.
70. x = –5
71. x = 1
73. x = –2
72. x = –3
74. Write a discussion of continuity for the function below. Be sure to include in your discussion where the
function is continuous, and, for the values where the function is not continuous, use the three part
definition to establish discontinuity.
Graph of g(x)
75. Write a discussion of continuity for the function below. Be sure to include in your discussion where the
function is continuous, and, for the values where the function is not continuous, explain why it is
discontinuous at that point.
Graph of h(x)
Find the value of a that makes each of the functions below everywhere continuous. Write the two limits that
must be equal in order for the function to be continuous.
2

4  x , x  1
76. f ( x)  
2

ax  1, x  1
2

 x  x  a, x  2
77. f ( x)  
3
2

ax  x , x  2
78. For what values of k and m is the function g(x) everywhere continuous? Use limits to set up your
work.
kx2  m,
x  2

 ln( 2 x  3)
g ( x)  e
, 2 x3
 kx  m,
x3

79. Show, using the intermediate value theorem, that the function F(x) = x3 + 2x – 1 has a zero on the
interval [0, 1]. CALCULATOR PERMITTED
80. Show, using the intermediate value theorem, whether or not the function g(θ) = θ2 – 2 – cosθ has a
zero on the interval [0, π]. CALCULATOR PERMITTED
For exercises 81 – 83, first, verify that the I.V.T. is applicable for the given function on the given interval.
Then, if it is applicable, find the value of the indicated c, guaranteed by the theorem.
81. f(x) = x2 – 6x + 8
[0, 3]
f(c) = 0
82. g(x) = x3 – x2 + x – 2
[0, 3]
f(c) = 4
x2  x
83. f ( x) 
x 1
52 ,4
f(c) = 6
For exercises 84 – 88, find each of the following limits at infinity. What do the results show about the existence
of a horizontal asymptote?
84.
87.
lim
x  
lim
x  
3x  2  5 x 2
2 x 2  3x  1
85. lim
3x  5
x  2x
2
86.
 3x
2x  1
x2  x
88. lim
x 
 2x 2  x
2x 2  3
 2x 2  5
x   3x  2
lim