AP Calculus AB
Unit 2 – Limits and Continuity
AP Calculus AB – Worksheet 7 Introduction to Limits
There are no great limits to growth because there are no limits of human intelligence, imagination, and wonder.
– Ronald Reagan
Answer the following questions.
1
For the function
( )
2
5,f x x=
as the x-value gets closer and closer to 3,
( )
fx
gets closer and closer to what
value?
2
For the function
( )
2
4
2
x
fx
x
−
=
−
, as the x-value gets closer and closer to 2,
( )
fx
gets closer and closer to what
value?
3
For the function
( )
1
x
f x e=+
, as the x-value gets closer and closer to 0,
( )
fx
gets closer and closer to what
value?
4
The graph of
( )
fx
is given below, use the graph to answer the following questions.
a)
( )
4
lim
x
fx
−
→
b)
( )
4
lim
x
fx
+
→
c)
d)
( )
4f
e)
( )
1
lim
x
fx
−
→
f)
( )
1
lim
x
fx
+
→
g)
h)
( )
1f
5
Use the graph of
( )
fx
to estimate the limits and value of the function, or explain why the limit does not exist.
a)
( )
4
lim
x
fx
−
→
b)
( )
4
lim
x
fx
+
→
c)
d)
( )
4f
e)
( )
2
lim
x
fx
−
→
f)
( )
2
lim
x
fx
+
→
g)
e)
( )
2f
6
Answer each statement as either True or False.
a) If
( )
15f =
, then
exists.
b) If
( )
15f =
and
exists, then
( )
1
lim 5
x
fx
→
=
.
c) If
( )
1
lim 5
x
fx
→
=
, then
( )
15f =
.
7
Use the graph of
( )
fx
above to answer each statement as either True or False.
a)
exists
b)
( )
0
lim 1
x
fx
→
=
c)
( )
0
lim 0
x
fx
→
=
d)
( )
1
lim 2
x
fx
→
=
e)
( )
1
lim 1
x
fx
→
=−
f)
exists for every
point c in
( )
3,1−
.
g)
does not exist
h)
i)
( )
01f =
j)
( )
11f =−
k)
( )
12f =
l)
exists
8
Simplify
x
2
+ 7x +12
x
2
−16
AP Calculus AB – Worksheet 8 Failing Limits; Properties of Limits
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits:
( ) ( )
lim and lim
x c x c
f x L g x M
→→
==
.
-
lim
xc
kk
→
=
-
lim
xc
xc
→
=
-
( ) ( ) ( ) ( )
lim lim lim
x c x c x c
f x g x f x g x L M
→ → →
= =

-
( ) ( ) ( ) ( )
lim lim lim
x c x c x c
f x g x f x g x L M
→ → →
= =

-
( )
( )
( )
( )
lim
lim
lim
xc
xc
xc
fx
fx
L
g x g x M
→
→
→
==
;
0M
-
( )
lim
xc
bf x bL
→
=

-
( )
0
lim
n
n
x
f x L
→
=
-
( )
0
lim
n
n
x
f x L
→
=

1) Suppose
( )
lim 5
xc
fx
→
=
,
( )
lim 2
xc
gx
→
=−
, and
( )
lim 9
xc
hx
→
=
, find
a)
( ) ( )
lim
xc
f x g x
→


b)
( ) ( )
lim
xc
f x g x
→
−

c)
( )
lim
xc
hx
→
d)
( )
1
lim
xc
gx
x
→
+


e)
( ) ( )
lim 2 3
xc
h x g x
→
−

f)
( )
( )
lim
xc
fx
hx
→




g)
( )
( )
lim
xc
gx
fx
→




h)
( )
2
lim
xc
gx
→


2) The graphs of f and g are given below. Use the graphs to evaluate each limit.
a)
( ) ( )
1
lim
x
f x g x
→−
+

b)
( )
( )
3
lim
x
f g x
→
c)
( ) ( )
3
lim
x
f x g x
→


d)
( ) ( )
2
lim 2 5
x
f x g x
→
+

e)
( )
( )
1
lim
x
fx
gx
→−




f)
( )
2
lim
x
xf x
→


3) Suppose that
( )
7
lim 0
x
fx
→
=
and
( )
7
lim 5
x
gx
→
=
. Determine the following limits.
a)
( )
( )
7
lim 5
x
gx
→
+
b)
( )
7
lim
x
xf x
→
c)
( )
2
7
lim
x
gx
→
d)
( )
( )
7
lim
1
x
gx
fx
→


−


AP Calculus AB – Worksheet 9 Evaluating Limits Algebraically
Once we accept our limits, we go beyond them. – Albert Einstein
Evaluate the limit.
1
( )
3
lim 3 2
x
x
→−
+
2
( )
32
1
lim 3 2 4
x
xx
→
−+
3
lim
x→3
x −1
x − 4
4
lim
x→2
cos
x
3
5
lim
x→1
sin
x
2
6
lim
x→0
sec 2x
7
2
3
9
lim
3
x
x
x
→
−
+
8
2
0
3 2 1
lim
x
xx
x
→
−+
9
2
1
5 6 10
lim
81
x
xx
x
→
++
−
10
0
2 5 sin
lim
5cos
x
xx
x
→
++
11
For the following piecewise function
( )
5 , 3
3
1, 3
4
xx
fx
x
x
−
=
+
, evaluate
.
For the following piecewise function:
( )
2
1, 2
1, 2 4
5, 4
xx
f x x x
xx
+
= −
+
, evaluate:
12) a)
( )
2
lim
x
fx
−
→
b)
( )
2
lim
x
fx
+
→
c)
d)
( )
2f
13) a)
( )
4
lim
x
fx
−
→
b)
( )
4
lim
x
fx
+
→
c)
d)
( )
4f
AP Calculus AB – Worksheet 10 Algebraic Methods for Finding Limits; Indeterminate Forms
Evaluate each limit
1)
( )
32
1
lim 12 1
x
xx
→
+−
2)
3)
2
4
54
lim
2
x
xx
x
→
++
+
4)
2
0
2
lim
x
xx
x
→
−
5)
4
4
lim
2
x
x
x
→
−
−
6)
( )
22
4
lim sin cos
x
xx
→
+
7)
2
1
23
lim
1
x
xx
x
→−
−−
+
8)
3
13 4
lim
3
x
x
x
→
+−
−
9)
3
21
lim
3
x
x
x
→−
+
+
10)
2
2
8
98
lim
64
x
yy
y
→−
++
−
11)
2
10
100
lim
10
x
x
x
→
−
−
12)
( )
2
0
9 81
lim
h
h
h
→
+−
13)
for
( )
2 1, 1
3, 1
xx
fx
xx
+
=
−
14) Suppose
( )
3
lim 5
x
fx
→
=−
and
( )
3
lim 2
x
gx
→
=
,
evaluate
( ) ( )
3
lim4 2
x
f x g x
→
−

15) Suppose
( )
2
,2
3 6, 2
x ax x
fx
xx
−
=
+
, find the value
a
that guarantees that
exists.
16) Suppose
( )
2
3
4 , 1
2, 1
x x x
fx
ax x
− −
=
− −
, find the value
a
that guarantees that
( )
1
lim
x
fx
→−
exists.
17) If
( )
7
9
lim 5
7
x
fx
x
→
+
=
−
, find
.
18) If
( )
4
7
lim 6
4
x
fx
x
→
−
=
−
, find
.
AP Calculus AB – Worksheet 11 Limits – The Difference Quotient/The Squeeze Theorem
The only limits to the possibilities in your life tomorrow are the “buts” you use today. – Les Brown
For #1-4, find
( ) ( )
0
lim
x
f x x f x
x
→
+ −
.
1.
( )
23f x x=+
2.
( )
2
4f x x x=−
3.
( )
4
fx
x
=
4.
( )
f x x=
Use the graph of
( )
fx
shown below to answer 5-7. The domain of
( )
fx
is
( )
,−
.
5.
6.
( )
3
lim
x
fx
→−
7. Identify the values of c for which
lim
x→c
f x
( )
exists. (Hint: think interval)
Find the limit, if it exists.
8.
lim
x→1
x
2
+ 3x + 2
x −1
9.
10.
lim
x→2
x − 3
x
2
− 25
11.
( )
2
6
lim sin
x
x
→
12
13
14
If
( )
2
2 2cosx g x x−
for all x, find
( )
0
lim
x
gx
→
.
15
If
( )
4
5
lim 1,
2
x
fx
x
→
−
=
−
find
.
AP Calculus AB – Worksheet 12 Limits at Infinity and Review
Review
For questions 1-2, use the graph of the function f shown above.
1
a)
( )
2
lim
x
fx
−
→
b)
( )
2
lim
x
fx
+
→
c)
d)
( )
2f
2
a)
( )
4
lim
x
fx
−
→
b)
( )
4
lim
x
fx
+
→
c)
d)
( )
4f
3
( )
2
4
lim 3 7
x
xx
→
− + =
4
3
1
lim 7 1
x
xx
→
+ + =
5
2
2
3
9
lim
2 15
x
x
xx
→−
−
=
−−
6
63
53
0
26
lim
43
x
xx
xx
→
+
=
+
7
3
3
lim
3
x
x
x
−
→
−
=
−
8
Given
( )
( )
2
2
sin , 2
18, 2
x x x
fx
x cx x
=
+ −
, find the value of c that
exists
Limits at Infinity
Evaluate each limit
1)
( )
32
lim 5 7 1
x
xx
→
− − +
2)
( )
32
lim 5 7 1
x
xx
→−
−+
3)
2
9
lim 8
x
x
→

+


4)
2
2
2 6 9
lim
x
xx
x
→
++
5)
98
lim
97
x
x
x
→
+
+
6)
98
lim
97
x
x
x
→−
+
+
7)
2
24
lim
3
x
x
x
→
−
+
8)
2
4
lim
3
x
x
x
→−
−
+
9)
3
3
lim
x
x
x
e
→
10)
3
3
lim
x
x
x
e
→−
11)
sin3
lim
15
x
x
x
→
12)
7 6 sin2
lim
6 cos2
x
xx
xx
→−
−+
+
13)
14) Determine the horizontal asymptote(s) of each of
the following.
a)
2
2
20
14
xx
y
x
−
=
+
b)
( )
( )( )
( )
2
3 8 5 4
21
xx
fx
x
+−
=
+
c)
( )
2
1
x
gx
x
=
−
AP Calculus AB - Worksheet 13 Continuity
Strive for continuous improvement, instead of perfection. – Kim Collins
3-Part Definition of Continuity
1. Determine the x-values at which the function f below has discontinuities. For each value state which condition is
violated from the 3-part definition of continuity.
I.
( )
fc
is defined
II.
exists.
III.
( ) ( )
lim
xc
f c f x
→
=
Show (THREE STEPS) that each of the following functions is either continuous or discontinuous at the given value of x.
2.
( )
5 at 1f x x x= + =
3.
( )
2
2 1 at 0f x x x x= + − =
4.
( )
2
16
at 4
4
x
f x x
x
−
==
−
5.
( )
2
25
at 5
5
x
f x x
x
−
==
+
6.
( )
1
at 3f x x
x
==
7.
( )
31
at 3
26
x
f x x
x
−
= = −
+
8. Determine the intervals on which the graph of
( )
fx
, shown above, is continuous
State the open interval(s) on which each function is continuous.
9.
( )
2
2f x x=+
10.
( )
1
fx
x
=
11.
( )
2
1
1
x
fx
x
+
=
−
12.
( )
2
2
32
45
xx
fx
xx
−+
=
+−
( )
2
1, 1 0
2 , 0 1
1, 1
2 4, 1 2
0, 2 3
xx
xx
f x x
xx
x
− −

==
− +

13. Given
( )
fx
and the graph of
( )
fx
, state why continuity fails at each value of x.
14. State the interval(s) on which
( )
fx
is continuous.
AP Calculus AB – Worksheet 14 Continuity - Removable
For questions 1 & 2,
a) Determine the x-coordinate of each discontinuity on the graph of
( )
fx
.
b) Identify each discontinuity as either removable or jump.
c) Evaluate the limit at each discontinuity.
d) State the interval(s) on which
( )
fx
is continuous.
1. 2.
For questions 3-5, answer the following:
a) Determine the x-coordinates of any discontinuities on the graph of
( )
fx
.
b) Identify the discontinuities as either infinite or removable.
c) Evaluate the limit at each removable discontinuity.
d) State the interval(s) on which
( )
fx
is continuous.
e) Write an extended function of
( )
gx
that is continuous.
3.
( )
2
2
56
x
fx
xx
−
=
−+
4.
( )
2
4
2
x
fx
x
−
=
+
5.
( )
2
2
12
68
xx
fx
xx
+−
=
++
6. Find the value of a that makes
( )
2
2 4, 2
3, 2
xx
fx
ax x
−
=
+
continuous on the entire interval.
7. Find the value of a that makes
( )
2
2, 3
4 1, 3
ax x
fx
xx
+
=
−
continuous on the entire interval.
8. Find the value of k that makes
( )
( )( )
2 1 2
,2
2
2
xx
x
fx
x
kx
+ −
=
−
=
continuous on the entire interval.
Evaluate each limit
9.
99
lim
x
x
x
e
→
10.
lim
ln
x
x
e
x
→
11.
99
ln
lim
x
x
x
→
12.
99
lim
x
x
x
e
→−
AP Calculus AB - Worksheet 15 Continuity – Infinite Discontinuities
Each of the following has a removable discontinuity. Find an extended function,
( )
gx
, that removes the discontinuity.
1)
( )
2
56
3
xx
fx
x
−+
=
−
2)
( )
2
5
5
x
fx
x
−
=
−
3)
( )
3
8
2
x
fx
x
+
=
+
4) Find each limit
a)
3
3
lim
3
x
x
x
−
→−
−
+
b)
3
3
lim
3
x
x
x
+
→−
−
+
5) Find the vertical asymptote on the graph of
( )
2
10
9
xx
fx
x
−
=
+
. Describe the behavior of
( )
fx
to the left and right
of the vertical asymptote.
For problems 6-7,
a) Determine the x-coordinate of the discontinuities on the graph of
( )
fx
. Identify the discontinuities as either
infinite or removable.
b) Use limits to describe the behavior of
( )
fx
near any removable discontinuities.
c) State the interval(s) on which
( )
fx
is continuous.
d) Identify any vertical asymptotes on the graph of
( )
fx
. Use limits to describe the behavior of
( )
fx
near the
vertical asymptote.
e) Use limits to identify any horizontal asymptotes on the graph of
( )
fx
.
f) Draw an accurate graph of
( )
fx
.
6)
( )
2
2
4
32
x
fx
xx
−
=
++
7)
( )
2
2
2 5 3
67
xx
fx
xx
−+
=
+−
8) Sketch the graph of any function such that:
( )
( )
( )
( )
1
lim
13
lim 2
lim 5
x
x
x
fx
f
fx
fx
→−
→
→
= −
=
=−
=
AP Calculus AB - Worksheet 16 Continuity 4
To live for results would be to sentence myself to continuous frustration. My only sure reward is in my actions and not
from them. – Hugh Prather
For problems 1-4, use the graph to test the function for continuity at the indicated value of x.
1.
1a) Explain why continuity fails at
1x=−
.
1b) What kind of discontinuity does
( )
fx
have?
1c) On what open interval(s) is
f x
( )
continuous?
2.
2a) Explain why continuity fails at
2x =
.
2b) What kind of discontinuity does
f x
( )
have?
2c) On what open interval(s) is
f x
( )
continuous?
3.
3a) Explain why continuity fails at
3x =
.
3b) What kind of discontinuity does
( )
fx
have at
3x =
?
3c) On what open interval(s) is
( )
fx
continuous?
4.
4a) Explain why continuity fails at
1x =
.
4b) What kind of discontinuity does
( )
fx
have?
4c) On what open interval(s) is
( )
fx
continuous?
Determine whether
f x
( )
is continuous at the given value of x.
5)
( )
2
sin , 2
, 2
3 9, 2
xx
f x x
x x x
==
+ −
6)
( )
2
2 3, 1
, 1
, 1
xx
f x x
xx
− +
==
Find the constant a, or the constants a and b, such that the function is continuous on the entire number line.
7)
( )
3
2
,2
,2
xx
fx
ax x
=
8)
( )
2
2
4 , 1
1, 1
xx
fx
ax x
− −
=
− −
9)
( )
2, 1
, 1 3
2, 3
x
f x ax b x
x
−
= + −
−
10)
( )
2
3 , 2
,2
7 4, 2
x x x
f x a x
xx
+
==
−
AP Calculus AB – Worksheet 17 Intermediate Value Theorem
In 1-3, verify that the Intermediate Value Theorem guarantees that there is a zero in the interval for the given function.
1)
( )
43
1
3
16
f x x x= − +
;
1,2
2)
( )
3
32f x x x= + −
;
2,1−
3)
( )
2
cosf x x x x= − −
;
0,
In 4-6, verify that the Intermediate Value Theorem applies to the indicated interval and find the value of c guaranteed
by the theorem. No calculator is permitted on these problems.
4)
( )
2
1f x x x= + −
( )
11 in 0,5fc=
5)
( )
2
68f x x x= − +
( )
5 in 0,3fc=
6)
( )
32
2f x x x x= − + −
( )
4 in 0,3 fc=
7
Given selected values of the continuous function
( )
hx
, what is the fewest number of times
( )
43hx=
in the
interval
0,30
? Explain your reasoning.
8
Water is pumped into a tank at a rate modeled by
( )
2
20
2000
t
W t e
−
=
liters per hour for
08t
, where t is measured
in hours. Water is removed from the same tank at a rate modeled by
( )
Rt
liters per hour, where
R
is continuous and
decreasing on
08t
. Selected values of
( )
Rt
are shown in the table below.
For
08t
, is there a time t when the rate at which water is pumped into the tank is the same as the rate at
which water is removed from the tank? Explain your reasoning.
9
For each of the graphs below state how continuity is destroyed at
x = c
.
10
Determine whether
( )
fx
is continuous at
x = −1
for
( )
1
, 1
1
, 1 1
2
, 1
x
x
x
f x x
xx
−
−
= −
AP Calculus AB - Worksheet 18 Limits and Their Properties Review
1
lim
x→
2
1+ sin x
1− cos x
2
lim
x→
cos 2x
x
2
3
lim
x→
6x
3
− 5x
x
2
+ 4x
3
4
lim
x→
x
2
+ x
4
x
2
+ x
6
5
lim
x→−
8x
3
− 5x
x
2
− 3x
6
lim
x→4
−
5
x − 4
7
lim
x→2
4x
3
− 32
5x
2
− 20
8
1
lim
1
x
x
e
→−
+
9
3
lim
43
x
x
e
x
→−
−
10
0
sin3
lim
7
x
x
x
→
11
Sketch a graph of a function such that:
( ) ( )
( ) ( )
( ) ( )
3
22
3 2 lim 1
lim lim
lim 5 lim
x
xx
xx
f f x
f x f x
f x f x
−+
→
→− →−
→− →
= = −
= − =
= = −
12
Find the value of a that will make
( )
gx
continuous.
( )
( )
2
3, 1
10, 1
a x x
gx
x a x
+
=
+ −
Use the following graph of
f x
( )
to answer the next five questions. If the limit does not exist, state why.
13
( )
1
lim
x
fx
+
→
14
( )
1
lim
x
fx
−
→
15
16
( )
1
lim
x
fx
→−
17
18
Evaluate
( ) ( )
0
lim
h
f x h f x
h
→
+−
for
( )
2
27f x x x=−
19
If
( )
2 1, 1
3 1, 1
xx
fx
xx
−
=
− +
, determine if
( )
fx
continuous at
x = 1.
20
If
( )
2
68
, 2
2
2, 2
xx
x
fx
x
x
++
−
=
+
=−
, determine if
( )
fx
is continuous at
x = −2
.
21
Find an extended function of
( )
2
16
4
x
fx
x
−
=
−
that is continuous.
Answers
Worksheet 8
1)
2)
a) 3
b) 2
c) 0
d) 8
e)
1
2
f) -2
3)
a) 10
b) 0
c) 25
d) -5
4) -9
5) 7
a) -10
b) 7
c) 3
d)
1
c
−
e) 24
f)
5
9
g)
2
5
−
h) 4
Worksheet 9
1) -7
2) 5
3)
− 2
4)
−
1
2
5) 1
6) 1
7) 0
8) DNE
9) 3
10)
2
5
11) DNE
12 ) a) 3
b) 3
c) 3
d) undefined
13 ) a) 15
b) 3
c) DNE
d) 3
Worksheet 10
1) 12
2)
6
7
3)
20
3
4)
2−
5) 4
6) 1
7) -5
8)
1
8
9) DNE
10)
7
16
11) 20
12) 18
13) DNE
14) -36
15)
4a =−
Worksheet 12
Quiz Review
1) 2; 2; 2; 1
2) 2; 4; DNE; 4
3) 11
4) 3
5)
3
4
6) 2
7) -1
8)
7
2
Limits at Infinity
1)
−
2)
3) 8
4) 9
5) 1
6) 1
7)
2
8)
2−
9)
0
10)
−
11) 0
12) -1
13) I and III
14)
a)
5y =
b)
3y =−
c)
1; 1yy= = −
Worksheet 18
1. 2
2. 0
3.
3
2
4. 0
5. DNE
6.
−
7.
12
5
8. 1
9. 0
10.
3
7
11. Answers
will vary
12. -4, 3
13. 1
14. -2
15. DNE
16. 2
17. 1
18.
47x−
19.
( )
( ) ( )
( )
_
11
1
I. 1 1
lim 1 lim 2
II.
limDNE
is not continuous at 1
xx
x
f
f x f x
f x x
+
→→
→
=

= = −


=
20.
( )
( )
( ) ( )
( )
2
22
2
I. 2 2
68
II. lim lim 4 2
2
III. 2 lim 2
is continuous at 2
xx
x
f
xx
x
x
f f x
f x x
→− →−
→−
−=
++
= + =
+
− = =
= −
21.
( )
is not continuous at 4f x x =
( )( )
( )
( )
( )
2
4 4 4
2
44
16
lim lim lim 4
44
16
,4
4
8 , 4
x x x
xx
x
x
xx
x
x
gx
x
x
→ → →
−+
−
= = +
−−
−
=
−
=