
Limits and Continuity 3-31
Example 3.28 Find the vertical asymptotes of the function
f x
( ) .=
+
−
Solution:
Now,
Also,
lim .
lim .
x
x
x
x
x
x
→
→
+
−
+
−
= ∞
+
−
=−∞
3
3
4
3
4
3
Hence, x = 3 is a vertical asymptote of (x + 4)/(x + 3).
Example 3.29 Find the vertical asymptotes of the function
f x
x x
x
( ) .=
+ +
+
2
3 2
2
Solution: Consider the function
lim
x
x x
x
→−
+ +
+
2
2
3 2
2
.
It is clear that
lim lim
( )( )
lim [ ] .
x x
x
x x
x
x x
x
x
→− →−
→−
+ +
+
=
+ +
+
= + = −
2
2
2
2
3 2
2
1 2
2
1 1
Hence, x = -2 is not a vertical asymptote. Therefore, f has no vertical asymptotes.
Example 3.30 Find the vertical asymptotes of the function
f x
x x
x x
( ) .=
+ +
− +
2
2
6 8
6 8
Solution: The concept of discontinuity ...