You are about to erase your work on this activity. Are you sure you want to do this?
Updated Version Available
There is an updated version of this activity. If you update to the most recent version of this activity, then your current progress on this activity will be erased. Regardless, your record of completion will remain. How would you like to proceed?
We explore functions that “shoot to infinity” near certain points in their
Consider the function
While the does not exist, something can still be said.
If grows arbitrarily large as approaches , we write and say that the limit of is
equal to infinity as goes to .
If grows arbitrarily large as approaches and is negative, we write and say that the
limit of is equal to negative infinity as goes to .
Which of the following are correct?
, so , so as ,
On the other hand, consider the function
While the two sides of the limit as approaches do not agree, we can still consider the
one-sided limits. We see and .
If at least one of the following hold:
then the line is a vertical asymptote of .
Find the vertical asymptotes of
Since is a rational function, it is continuous on its domain. So the only points where
the function can possibly have a vertical asymptote are zeros of the denominator.
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
Consider the one-sided limits separately. Since approaches from the right and the
numerator is negative, . Since approaches from the left and the numerator is