How do you tell if a graph of a rational function has a hole in the graph or a vertical asymptote? Give an example of each case.

Answer 1

To determine what is what in a rational function, the first step is to factor it. Once you factor it, a factor that cancels because it is in both the numerator AND denominator is a hole. If it doesn't cancel, it is a vertical asymptote.

Example 1:

(x+2)(x+3)

(x-4)(x+3)

(x+3) is in both the numerator and denominator, therefore there is a hole at the zero for this factor, x=-3. The other factor, there is a vertical asymptote at the zero, x=4.

Example 2:

(x+2)(x+3)

(x-4)(x-2)

There are no common factors in both the numerator and denominator, therefore there are no holes. The remaining factors in the denominator are both vertical asymptotes. There is one at x=4 and a second at x=2.