CALCULUS
Limits. Functions defined by a graph
1. Consider the following function defined by its graph:
e
y6
3
u
e
e
2
u
1
x
-
0
−5
−4
−3
−2
−1
0
−1
1
2
3
4
−2
e
u
−3
e
5
−4
Find the following limits:
a) lim − f (x)
x→−1
b) lim + f (x)
x→−1
c) lim f (x)
d) lim f (x)
x→−1
e) lim f (x)
x→−4
x→4
2. Consider the following function defined by its graph:
y6
e
3
u
e
2
u
e
1
e
0
−5
−4
−3
−2
−1
0
−1
1
2
3
4
e
u
5
x
-
−2
−3
−4
Find the following limits:
a) lim f (x)
x→1−
b) lim f (x)
x→1+
c 2009 La Citadelle
c) lim f (x)
x→1
d) lim f (x)
x→−5
1 of 9
e) lim f (x)
x→5
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CALCULUS
Limits. Functions defined by a graph
3. Consider the following function defined by its graph:
y6
e
3
u
e
2
1
x
-
0
−5
−4
−3
−2
−1
u
e
0
−1
1
2
3
−2
e
−3
u
4
5
−4
Find the following limits:
a) lim f (x)
x→−2−
b) lim f (x)
x→−2+
c) lim f (x)
d) lim f (x)
x→−2
e) lim f (x)
x→−4
x→3
4. Consider the following function defined by its graph:
y6
3
−5
e
−4
u
2
e
1
u
x
-
0
−3
−2
−1
0
−1
1
2
3
4
5
e
u
−2
−3
−4
Find the following limits:
a) lim− f (x)
x→2
b) lim+ f (x)
x→2
c 2009 La Citadelle
c) lim f (x)
x→2
d) lim f (x)
x→−4
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e) lim f (x)
x→5
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CALCULUS
Limits. Functions defined by a graph
5. Consider the following function defined by its graph:
y6
3
2
1
e
−5
u
e
e
0
−4
−3
−2
e
−1
0
−1
1
2
3
4
u
5
x
-
−2
u
−3
−4
Find the following limits:
a) lim f (x)
x→−2−
b) lim f (x)
x→−2+
c) lim f (x)
d) lim f (x)
x→−2
e) lim f (x)
x→−5
x→5
6. Consider the following function defined by its graph:
e
e
u
y6
u
e
3
e
2
u
e
1
x
-
0
−5
−4
−3
−2
−1
0
−1
1
2
3
4
5
−2
−3
−4
Find the following limits:
a) lim − f (x)
x→−2
b) lim + f (x)
x→−2
c 2009 La Citadelle
c) lim f (x)
x→−2
d) lim f (x)
x→−5
3 of 9
e) lim f (x)
x→5
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CALCULUS
Limits. Functions defined by a graph
7. Consider the following function defined by its graph:
y6
3
u
e
2
−5
e
1
e
u
−4
0
−3
e
u
−2
−1
0
−1
x
1
2
3
4
5
−2
−3
−4
Find the following limits:
a) lim f (x)
x→−2−
b) lim f (x)
x→−2+
c) lim f (x)
d) lim f (x)
x→−2
e) lim f (x)
x→−4
x→5
8. Consider the following function defined by its graph:
y6
3
2
e
u
−5
e
−3
−4
e
1
e
u
−2
0
−1
0
−1
x
1
2
3
4
5
e
u
−2
−3
−4
Find the following limits:
a) lim − f (x)
x→−2
b) lim + f (x)
x→−2
c 2009 La Citadelle
c) lim f (x)
x→−2
d) lim f (x)
x→−3
4 of 9
e) lim f (x)
x→5
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CALCULUS
Limits. Functions defined by a graph
9. Consider the following function defined by its graph:
y6
3
2
1
−5
x
-
0
u
e
−4
e
−3
−2
−1
0
−1
1
2
u
e
3
−2
4
e
5
u
−3
−4
Find the following limits:
a) lim f (x)
x→2−
b) lim f (x)
c) lim f (x)
x→2
x→2+
d) lim f (x)
e) lim f (x)
x→−4
x→4
10. Consider the following function defined by its graph:
y6
e
3
u
2
1
x
-
0
−5
−4
−3
u
e
−2
e
−1
0
−1
1
2
3
4
5
−2
−3
u
e
−4
Find the following limits:
a) lim − f (x)
x→−2
b) lim + f (x)
x→−2
c 2009 La Citadelle
c) lim f (x)
x→−2
d) lim f (x)
x→−3
5 of 9
e) lim f (x)
x→3
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Answers:
2. a) 3
3. a) − 2
4. a) 2
5. a) − 1
6. a) 3
7. a) 0
8. a) 0
9. a) − 1
10. a) 3
c 2009 La Citadelle
1. a) 2
b) 2
c) 2
d) DN E
e) DN E
b) 3
c) 3
d) DN E
e) DN E
b) − 2
c) − 2
d) DN E
e) − 2
b) 2
c) 2
d) 0
e) − 1
b) − 1
c) − 1
d) DN E
e) 1
b) 4
c) DN E
d) DN E
e) 2
b) 0
c) 0
d) DN E
e) 3
b) 1
c) DN E
d) DN E
e) − 1
b) − 1
c) − 1
d) DN E
e) − 1
b) 3
c) 3
d) DN E
e) − 3
CALCULUS
Limits. Functions defined by a graph
CALCULUS
Limits. Functions defined by a graph
Solutions:
1.
a)
b)
c)
d)
lim f (x) = 2
x→−1−
lim f (x) = 2
x→−1+
lim f (x) =
x→−1−
lim f (x) 6=
x→−4−
lim f (x) . Therefore lim f (x) = 2
x→−1+
x→−1
lim f (x) . Therefore lim f (x) = DNE
x→−4+
x→−4
e) lim− f (x) 6= lim+ f (x) . Therefore lim f (x) = DNE
x→4
x→4
x→4
2.
a) lim− f (x) = 3
x→1
b) lim+ f (x) = 3
x→1
c) lim f (x) = lim f (x) . Therefore lim f (x) = 3
x→1−
d)
x→1+
lim − f (x) 6=
x→−5
x→1
lim + f (x) . Therefore lim f (x) = DNE
x→−5
x→−5
e) lim f (x) 6= lim f (x) . Therefore lim f (x) = DNE
x→5−
3.
a)
b)
c)
d)
x→5+
x→5
lim f (x) = −2
x→−2−
lim f (x) = −2
x→−2+
lim f (x) =
x→−2−
lim − f (x) 6=
x→−4
lim f (x) . Therefore lim f (x) = −2
x→−2+
x→−2
lim + f (x) . Therefore lim f (x) = DNE
x→−4
x→−4
e) lim− f (x) = lim+ f (x) . Therefore lim f (x) = −2
x→3
x→3
x→3
4.
a) lim− f (x) = 2
x→2
b) lim+ f (x) = 2
x→2
c) lim f (x) = lim f (x) . Therefore lim f (x) = 2
x→2−
d)
x→2+
lim − f (x) =
x→−4
x→2
lim + f (x) . Therefore lim f (x) = 0
x→−4
x→−4
e) lim f (x) = lim f (x) . Therefore lim f (x) = −1
x→5−
5.
a)
b)
c)
d)
x→5+
x→5
lim f (x) = −1
x→−2−
lim f (x) = −1
x→−2+
lim f (x) =
x→−2−
lim − f (x) 6=
x→−5
lim f (x) . Therefore lim f (x) = −1
x→−2+
x→−2
lim + f (x) . Therefore lim f (x) = DNE
x→−5
x→−5
e) lim f (x) = lim f (x) . Therefore lim f (x) = 1
x→5−
6.
a)
b)
c)
d)
x→5+
x→5
lim f (x) = 3
x→−2−
lim f (x) = 4
x→−2+
lim f (x) 6=
x→−2−
lim − f (x) 6=
x→−5
lim f (x) . Therefore lim f (x) = DNE
x→−2+
x→−2
lim + f (x) . Therefore lim f (x) = DNE
x→−5
x→−5
e) lim− f (x) = lim+ f (x) . Therefore lim f (x) = 2
x→5
x→5
c 2009 La Citadelle
x→5
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CALCULUS
7.
a)
b)
c)
d)
Limits. Functions defined by a graph
lim f (x) = 0
x→−2−
lim f (x) = 0
x→−2+
lim f (x) =
x→−2−
lim − f (x) 6=
x→−4
lim f (x) . Therefore lim f (x) = 0
x→−2+
x→−2
lim + f (x) . Therefore lim f (x) = DNE
x→−4
x→−4
e) lim− f (x) = lim+ f (x) . Therefore lim f (x) = 3
x→5
8.
a)
b)
c)
d)
x→5
x→5
lim f (x) = 0
x→−2−
lim f (x) = 1
x→−2+
lim f (x) 6=
x→−2−
lim − f (x) 6=
x→−3
lim f (x) . Therefore lim f (x) = DNE
x→−2+
x→−2
lim + f (x) . Therefore lim f (x) = DNE
x→−3
x→−3
e) lim− f (x) = lim+ f (x) . Therefore lim f (x) = −1
x→5
x→5
x→5
9.
a) lim− f (x) = −1
x→2
b) lim f (x) = −1
x→2+
c) lim− f (x) = lim+ f (x) . Therefore lim f (x) = −1
x→2
d)
x→2
lim − f (x) 6=
x→−4
x→2
lim + f (x) . Therefore lim f (x) = DNE
x→−4
x→−4
e) lim− f (x) = lim+ f (x) . Therefore lim f (x) = −1
x→4
x→4
x→4
10.
a) lim f (x) = 3
x→−2−
b)
c)
d)
lim f (x) = 3
x→−2+
lim f (x) =
x→−2−
lim − f (x) 6=
x→−3
lim f (x) . Therefore lim f (x) = 3
x→−2+
x→−2
lim + f (x) . Therefore lim f (x) = DNE
x→−3
x→−3
e) lim− f (x) = lim+ f (x) . Therefore lim f (x) = −3
x→3
x→3
c 2009 La Citadelle
x→3
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