Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Determine whether each function is continuous at the given -values. Justify using the continuity test. If
discontinuous, identify the type of discontinuity as infinite, jump, or removable.
1.
2.
>
=
+ −=
=
7
−
=
<
y
5
y
6
4
5
4
3
3
2
2
1
1
-7 -6 -5 -4 -3 -2 -1 -1
x
x
-5
1 2 3 4 5 6 7
-4
-3
-2
-1
1
-1
-2
-2
-3
-4
-3
-5
-4
-6
-5
Copyright © PreCalculusCoach.com
1
2
3
4
5
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
3.
+
=
4.
≥
−
<
5
=
=
+=
10 y
y
8
4
3
6
2
4
2
1
x
x
-5
-4
-3
-2
-1
1
2
3
4
-8
5
-6
-4
-2
2
-1
-2
-2
-4
-3
-6
-4
-8
-5
-10
Copyright © PreCalculusCoach.com
2
4
6
8
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
5.
+
5
=
y
−
6.
=
−
12
=
=
−
y
= − =
+
4
10
8
3
6
2
4
1
2
x
-5
-4
-3
-2
-1
1
234
5
-14 -12 -10 -8 -6 -4 -2
-2
-1
-4
-2
-6
-3
-8
-4
-10
-12
-5
Copyright © PreCalculusCoach.com
3
x
2 4
6 8 10 12 14
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Find the value of so that
7.
+
is continuous.
8.
>
+
>
=
+
≤
=
=
+
<
Determine between which consecutive integers the real zeros of function are located on the given interval.
9.
=
+ −
5
y
10.
[ , ]
−
=
5
−
[− , ]
4
3
4
3
2
2
1
y
1
x
-5
-4
-3 -2 -1
1234
x
5
-5
-4
-3
-2 -1
1234
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
Copyright © PreCalculusCoach.com
4
5
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Use the graph of each function to describe its end behavior. Support the conjecture numerically.
=−
11.
−
+
5
+
y
4
3
2
1
x
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-3
-4
-5
= −+
12.
14
y
12
10
8
6
4
2
-14 -12 -10 -8 -6 -4 -2 -2
x
2
4
6
8 10 12 14
-4
-6
-8
-10
-12
-14
Evaluate the following limits.
13.
14.
−
→
=
−
=?
Copyright © PreCalculusCoach.com
5
→ − + + =?
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing.
15.
=− + +
y
5
4
3
2
1
x
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-3
-4
-5
16.
=+ −
y
16
12
8
4
x
-16
-12
-8
-4
4
8
12
16
-4
-8
-12
-16
Copyright © PreCalculusCoach.com
6
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
ANSWERS
Determine whether each function is continuous at the given -values. Justify using the continuity test. If
discontinuous, identify the type of discontinuity as infinite, jump, or removable.
1.
2.
=
>
+ −=
=
7
−
=
<
y
y
5
6
4
5
4
3
3
2
2
1
1
x
x
-7 -6 -5 -4 -3 -2 -1 -1
-5
2 3 4 5 6 7
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-2
-3
-4
-3
-5
-4
-6
-5
= + −
=
>
=
=
−
<
= + ∗ −
=
>
=−
=
−
<
=
→
→−
F
→
→
−
+
I
.
.
− .
− .
− →
→ F
.
− .
−
→
.
.
− .
− .
=−
−
=
( )
→
.
I
− .
− .
− .
− .
− .
.
.
.
.
.
− .
→
− .
.
=
→
=
+
−
=
>
=
−
<
→
=
=
Copyright © PreCalculusCoach.com
7
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
3.
+
=
4.
≥
−
=
<
5
=
+=
10 y
y
8
4
3
6
2
4
2
1
x
x
-5
-4
-3
-2
-1
1
2
3
4
-8
5
-6
-4
-2
2
-1
-2
-2
-4
-3
-6
-4
-8
-5
-10
+
≥
=
+
=
+
4
6
8
=
=
=−
<
= + = ∗ + =
=
→
→ F
→
− →
F
− .
− .
− .
− .
I
.
.
.
.
.
.
.
.
.
.
.
− .
→
→
.
− .
I
→
+ →
.
.
.
.
.
.
=
=
=
→
→
+
=
Copyright © PreCalculusCoach.com
8
= ( )
+
=
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
5.
+
y
5
=
−
6.
=
−
y
12
=
=
=
−
−
=
+
4
10
8
3
6
2
4
1
2
x
-5
-4
-3
-2
-1
1
23
4
5
x
-14 -12 -10 -8 -6 -4 -2
2
4
6 8 10 12 14
-2
-1
-4
-2
-6
+
=
-3
-8
-4
-10
-5
-12
= −
−
−
−
=
−
=
+
=
−
=−
=
+
− +
=
=
− =
−
−
=
−
→− F
→≈− .
− .
− .
− .
− .
→−
− .
− .
− .
− .
→≈ − .
I
→−
− .
− .
− .
− .
( )≈− .
→F
→∞
→ F
I
− .
− .
− .
− .
− .
− .
− .
− .
− .
→F
− .
→−
I
− .
− .
→
=
+
F
→−
=
+
−
=− − =
→−
−
( )=−
→
→ F
.
.
.
.
.
.
−
−
−
− .
− .
− .
.
.
.
.
− ,
.
.
.
→
→∞
.
.
I
→
( )
→
( )≈ − .
but
→
− is undefined
( ) ==
→F
has a removable discontinuity at = −
has an infinite discontinuity at =
→F
but
( )
has a
removable discontinuity
−
is undefined
at
= −
→
( )=
=
=
Copyright © PreCalculusCoach.com
9
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Find the value of so that
+
7.
is continuous.
>
+
8.
>
=
=
+
∗
+
+
=
+
=
=
=
≤
+
=
+
∗
+
<
=
+
=
+ =
∗
+
+
=
=
∗+
=
Determine between which consecutive integers the real zeros of function are located on the given interval.
9.
=
+ −
5
10.
[ , ]
y
−
=
5
−
[− , ]
4
3
4
3
2
2
1
y
1
x
-5
-4
-3 -2 -1
1234
x
5
-5
-4
-3
-2 -1
-1
-1
-2
-2
-3
-3
-4
-4
-5
-5
−
− .
1234
− .
− .
.
is negative positive and
5
−
−
.
−
− is negative and
is positive,
− is positive,
− ≤ ≤−
is positive and
≤ ≤
has zero in interval:
is negative
≤ ≤
≤
≤
has zeros in intervals:
−
Copyright © PreCalculusCoach.com
10
≤
≤
−
≤
≤
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Use the graph of each function to describe its end behavior. Support the conjecture numerically.
=−
11.
−
+
5
+
y
4
From the graph, it appears that:
3
→ ∞ as → −∞ and
conjecture.
→ −∞ as → ∞ The table supports this
2
1
−
−
x
-5
-4
-3
-2
-1
1
2
3
4
-
5
∗
∗
-
∗
-1
-2
-3
-4
-5
= −+
12.
14
y
12
10
8
6
From the graph, it appears that:
4
2
-14 -12 -10 -8 -6 -4 -2 -2
→ as → −∞ and
conjecture.
x
2
4
6
→ as → ∞ The table supports this
8 10 12 14
−
-4
−
-6
-8
−
-10
-12
-14
Evaluate the following limits.
13.
14.
−
→
=
−
=?
Copyright © PreCalculusCoach.com
11
→ − + + =?
∗
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
−
−+
→
−
+ + =− + ∗ +
→
−
+ + =− + +
→
=
−
=
−
−
−
→
→
=
( → + )= + =
→
− + + =
+ =
Determine the interval(s) on which the function is increasing and the interval(s) on which the function is decreasing.
15.
=− + +
From the graph, it appears that:
5 y
A function
A function
The table
4
3
2
−
−
is increasing for
is decreasing for <
+ +
+ +
.
> .
supports this conjecture.
1
-5 -4 -3 -2 -1 -1
12345
−
x
.
.
-2
-3
-4
-5
16.
=+ −
From the graph, it appears that:
y
16
A function
12
+
−
A functio n
is increasing for < − .
for
is decreas ing
8
+
−
− . < < .
4
-16 -12-8
-4
A function
x
4
8
1216
The table
+
−
s increasing for >
.
supports this conjecture.
-4
−
-8
.
-12
-16
Copyright © PreCalculusCoach.com
− .
12
.
− . −
Name: _________________________________________________ Period: ___________ Date: ________________
Continuity, End Behavior, and Limits Assignment
Copyright © PreCalculusCoach.com
13