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