
3-36 Calculus – Differentiation and Integration
= +
=
→+∞
→+∞
lim
lim
/
x
x
x
x
x
x
x
x
x
4
2
4
4
4
4
1
2
1
+
=
→+∞
x
x
x
x x
1
4
1
2
1
1
/
/
lim ( )
+
=
+
→+∞ →+∞
x
x
x x
x
1
4
1
2
1
1/
lim lim
= +
1
4 0 1
/ x
(by the property (c) of the Theorem 3.2)
(( )
=
( )
=
=
∞
∞0
1
2
0 5 0
4
.
.
and
lim lim lim
/
x
x x
x
x x
x
→−∞ →−∞ →−∞
+
=
+
2 2 4
1
2
1
2
1
= ∞+
( )
= ∞
1
0
0
4 1
/
,
x
which is still an indeterminate form.
Therefore, the line y = 4 is a horizontal asymptote for the given function.
Example 3.37 Find the horizontal asymptotes of the function f x
x
x
( ) .=
+
+
4 2
3 1
2
Solution: We have