Part II: The Answers
220
lim
lim
lim
x
x
x
x
x
x
x
x
x
x
→
→
→
−
−
−
− −
−
=
−
(
) +
(
)
−
=
+
(
)
= +
=
3
2
3
3
6
3
3
2
3
2
3 2
5
And the right-hand limit at a is
lim
lim
lim
x
x
x
x
x
x
x
x
x
x
→
→
→
+
+
+
− −
−
=
−
(
) +
(
)
−
=
+
(
)
= +
=
3
2
3
3
6
3
3
2
3
2
3 2
5
Note that in this case, you could have simply evaluated the limit as x approaches 3
instead of examining the left-hand limit and right-hand limit separately.
The left-hand and right-hand limits match, so the limit exists and is equal to 5. Because
f
 
(a) = f
 
(3) = 5, the definition of continuity is satisfied and the function is continuous.
259.
continuous, f a
( ) = 1
8
A function f
 
(x) is continuous at x = a if it satisfies the equation lim x a f x
f a
→
( ) = ( ).
The left-hand limit at a is
lim
lim
lim
x
x
x
x
x
x
x
x
x
→
→
→
−
−
−
−
−
=
−
−
(
) +
(
)
=
+
(
)
= +
16
16
16
4
16
4
4
4
1
4
1
4
16
= = 1
8
The right-hand limit at a is
lim
lim
lim
x
x
x
x
x
x
x
x
x
→
→
→
+
+
+
−
−
=
−
−
(
) +
(
)
=
+
(
)
= +
16
16
16
4
16
4
4
4
1
4
1
4
16
= = 1
8
The left-hand and right-hand limits match, so the limit exists.
Note that in this case, you could have simply evaluated the limit as x approaches 16
instead of examining the left-hand limit and right-hand limit separately.
The value at a is given by f a
f
( ) = ( )=
16
1
8
, which matches the limit. Because the definition
of continuity is satisfied, you can conclude that that function is continuous at a = 16.
Answers
201–300