Infinite Limits and Asymptotes
Read this section to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1-8.
The Limit is Infinite
The function is undefined at , but we can still ask about the behavior of for values of "close to" . Fig. 6 indicates that if is very small, close to , then is very large. As the values of get closer to , the values of grow larger and can be made as large as we want by picking to be close enough to . Even though the values of are not approaching any number, we use the "infinity" notation to indicate that the values of are growing without bound, and write .
Fig. 6
The values of do not equal "infinity:" means that the values of can be made arbitrarily large by picking values of very close to .
The limit, as , of is slightly more complicated. If is close to , then the value of can be a large positive number or a large negative number, depending on the sign of .
The function does not have a (two-sided) limit as approaches , but we can still ask about one-sided limits:
Solution: (a) As , then and . Since the denominator is approaching we cannot use the Main Limit Theorem, and we need to examine the functions more carefully. If , then so . If is close to and slightly larger than , then the ratio of to is the ratio . As gets closer to is . By taking closer to , the denominator gets closer to but is always positive, so the ratio gets arbitrarily large and negative: .
(b) As , then and gets arbitrarily close to , and is negative. The value of the ratio is : .