Answers and Explanations 227
Answers
201–300
278.
2x
Use the derivative definition with f
 
(x) = x
2
and f
 
(x + h) = (x + h)
2
= (x + h)(x + h) =
x
2
+ 2xh + h
2
:
lim
lim
lim
lim
h
h
h
h
x
xh h x
h
xh h
h
h x h
h
x
→
→
→
→
+
+ −
=
+
=
+
(
)
=
+
0
2
2
2
0
2
0
0
2
2
2
2
h h
x
x
(
)
=
+
=
2
0
2
279.
1
Use the derivative definition with f
 
(x) = 2 + x and f
 
(x + h) = 2 + (x + h):
lim
lim
h
h
x h
x
h
h
h
→
→
+ + − +
(
)
=
=
0
0
2
2
1
280.
− 1
2
x
Use the derivative definition with f x
x
( ) = 1 and with f x h
x h
+
(
)= +
1 to get the
following:
lim
lim
lim
lim
h
h
h
h
x h x
h
x
x h
x x h
h
h
hx x h
→
→
→
→
+
−
=
− +
(
)
+
(
)
=
−
+
(
)
=
0
0
0
1
1
1
0 0
2
1
1
0
1
−
+
(
)
=
−
+
(
)
= −
x x h
x x
x