We explore functions that “shoot to infinity” near certain points in their domain.
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 .
On the other hand, consider the function
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 Now 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 negative, .
Hence we have a vertical asymptote at .
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. Since approaches from the right and the numerator is negative, . Since approaches from the left and the numerator is negative, .