We explore functions that “shoot to infinity” near certain points.
Consider the function
If grows arbitrarily large as approaches and is negative near , we write and say that the limit of is equal to negative infinity as goes to .
On the other hand, consider the function
Start by factoring both the numerator and the denominator: Using limits, we must investigate what happens with when and , since and are the only zeros of the denominator. Write Now write Consider the one-sided limits separately.
When , the quantity is positive and approaches and the numerator is negative, therefore, .
On the other hand, when , the quantity is negative and approaches and the numerator is negative, therefore, .