Answers and Explanations 213
245.
–∞
Notice that if you consider the limit immediately, you have the indeterminate form ∞ – ∞.
Begin by factoring to get
lim
lim
lim
lim
x
x
x
x
x x
x
x
x
x
→∞
→∞
→∞
→∞
−
(
)
=
−
(
)
= (
)
−
(
)
(
)
= ∞ −
1 2
1 2
1 2
1 2
1
1
∞ ∞
( )
= −∞
246.
3
2
Notice that if you consider the limit immediately, you have the indeterminate form ∞ – ∞.
Create a fraction and multiply the numerator and denominator by the conjugate of the
expression x
x
x
4
2
2
3
+
−
(
) . The conjugate is x x x
4
2
2
3
+
+
(
) , so you have the following:
lim
lim
x
x
x
x
x
x
x
x
x
x
x
x
x
x
→∞
→∞
+
−
(
)
=
+
−
(
) + +
+
+




4
2
2
4
2
2
4
2
2
4
2
2
3
3
1
3
3
 

=
+
+
→∞
lim
x
x
x
x
x
3
3
2
4
2
2
Next, multiply the numerator and denominator by 1
2
x
so you can simplify the
expression underneath the radical:
lim
lim
l
x
x
x
x
x
x
x
x
x
x
x
x
x
→∞
→∞
( )
+
+
(
)
=
+
+






=
1 3
1
3
3
1
3
2
2
2
4
2
2
2
4
2
2
2
i im
lim
li
x
x
x
x
x
x
x
x
x
x
x
→∞
→∞
+
(
) +






=
+
+






=
3
1
3
3
3
1
4
4
2
2
2
4
4
2
4
m m
x
x
→∞
+
+






3
1 3 1
2
Now apply the limit:
=
+ +
(
)
=
3
1 0 1
3
2
Answers
201–300