Chapter 3
THE DERIVATIVE
3.1 Limits
1.
Since
lim
x
!
2
¡
f
(
x
)
does not equal
lim
x
!
2
+
f
(
x
)
,
lim
x
!
2
f
(
x
)
does not exist. The answer is c.
2.
Since
lim
x
!
2
¡
f
(
x
) = lim
x
!
2
+
f
(
x
)=
¡
1
;
lim
x
!
2
f
(
x
)=
¡
1
.Theanswerisa
.
3.
Since
lim
x
!
4
¡
f
(
x
) = lim
x
!
4
+
f
(
x
)=6
,
lim
x
!
4
f
(
x
)=6
. The answer is b.
4.
Since
lim
x
!
1
¡
f
(
x
) = lim
x
!
1
+
f
(
x
)=
¡1
;
lim
x
!
1
f
(
x
)=
¡1
. The answer is b.
5. (a)
By reading the graph, as
x
gets closer to 3
from the left or right,
f
(
x
)
gets closer to 3.
lim
x
!
3
f
(
x
)=3
(b)
By reading the graph, as
x
gets closer to 0
from the left or right,
f
(
x
)
gets closer to 1.
lim
x
!
0
f
(
x
)=1
6. (a)
By reading the graph, as
x
gets closer to 2
from the left or right,
F
(
x
)
gets closer to 4.
lim
x
!
2
F
(
x
)=4
(b)
By reading the graph, as
x
gets closer to
¡
1
from left or right,
F
(
x
)
gets closer to 4.
lim
x
!¡
1
F
(
x
)=4
7. (a)
By reading the graph, as
x
gets closer to 0
from the left or right,
f
(
x
)
gets closer to 0.
lim
x
!
0
f
(
x
)=0
(b)
By reading the graph, as
x
gets closer to
2
from the left,
f
(
x
)
gets closer to
¡
2
, but as
x
gets
closer to 2 from the right,
f
(
x
)
gets closer to 1.
lim
x
!
2
f
(
x
)
does not exist.
8. (a)
By reading the graph, as
x
gets closer to 3
from the left or right,
g
(
x
)
gets closer to 2.
lim
x
!
3
g
(
x
)=2
(b)
By reading the graph, as
x
gets closer to
5
from the left,
g
(
x
)
gets closer to
¡
2
, but as
x
gets
closer to 5 from the right,
g
(
x
)
gets closer to 1.
lim
x
!
5
g
(
x
)
does not exist.
9. (a)
(i)
By reading the graph, as
x
gets closer to
¡
2
from the left,
f
(
x
)
gets closer to
¡
1
:
lim
x
!¡
2
¡
f
(
x
)=
¡
1
(ii)
By reading the graph, as
x
gets closer to
¡
2
from the right,
f
(
x
)
gets closer to
¡
1
2
:
lim
x
!¡
2
+
f
(
x
)=
¡
1
2
(iii)
Since
lim
x
!¡
2
¡
f
(
x
)=
¡
1
and
lim
x
!¡
2
+
f
(
x
)
=
¡
1
2
;
lim
x
!¡
2
f
(
x
)
does not exist.
(iv)
f
(
¡
2)
does not exist since there is no
point on the graph with an
x
-coordinate
of
¡
2
:
(b)
(i)
By reading the graph, as
x
gets closer to
¡
1
from the left,
f
(
x
)
gets closer to
¡
1
2
:
lim
x
!¡
1
¡
f
(
x
)=
¡
1
2
(ii)
By reading the graph, as
x
gets closer to
¡
1
from the right,
f
(
x
)
gets closer to
¡
1
2
:
lim
x
!¡
1
+
f
(
x
)=
¡
1
2
(iii)
Since
lim
x
!¡
1
¡
f
(
x
)=
¡
1
2
and
lim
x
!¡
1
+
f
(
x
)=
¡
1
2
;
lim
x
!¡
1
f
(
x
)=
¡
1
2
:
(iv)
f
(
¡
1) =
¡
1
2
since
¡
¡
1
;
¡
1
2
¢
is a point
of the graph.