Answers and Explanations 215
248.
y = 3
5
To find any horizontal asymptotes of the function y
x
x
x
= +
+
1 3
5
4
4
, you need to
consider the limit of the function as x → ∞ and as x → –∞. For the limit as x → ∞, begin
by multiplying the numerator and denominator by 1
4
x
:
lim
lim
lim
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
→∞
→∞
→∞
+
+
=
+
+
=
+
+
1 3
5
1
3
5
1 3
1 5
4
4
4
4
4
4
4
4
4
3
= = +
+
=
0 3
0 5
3
5
To find the limit as x → –∞, you can proceed in the same way in order to simplify:
lim
lim
lim
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
→−∞
→ −∞
→−∞
+
+
=
+
+
=
+
1 3
5
1
3
5
1 3
1
4
4
4
4
4
4
4
4
4
3 3
5
0 3
0 5
3
5
+
= +
+
=
Therefore, the only horizontal asymptote is y = 3
5
.
249.
y = –1
To find any horizontal asymptotes of the function y
x
x
= −
+
5
5
2
2
, you need to
consider the limit of the function as x → ∞ and as x → –∞. For the limit as x → ∞, begin
by multiplying the numerator and denominator by 1
2
x
:
lim
lim
lim
x
x
x
x
x
x
x
x
x
x
x
x
x
→∞
→∞
→∞
−
+
=
−
+
=
−
+
= −
5
5
5
5
5 1
5 1
0 1
2
2
2
2
2
2
2
2
2
2
0 0 1
1
+
= −
Answers
201–300