MA001: College Algebra

Occasionally, a graph will contain a hole: a single point where the graph is not defined, indicated by an open circle. We call such a hole a removable discontinuity.

For example, the function $f(x)=\frac{x^{2}-1}{x^{2}-2 x-3}$ may be re-written by factoring the numerator and the denominator.

$f(x)=\frac{(x+1)(x-1)}{(x+1)(x-3)}$

Notice that $x+1$ is a common factor to the numerator and the denominator. The zero of this factor, $x=−1$, is the location of the removable discontinuity. Notice also that $x–3$ is not a factor in both the numerator and denominator. The zero of this factor, $x=3$, is the vertical asymptote. See Figure 10. [Note that removable discontinuities may not be visible when we use a graphing calculator, depending upon the window selected.

Figure 10

REMOVABLE DISCONTINUITIES OF RATIONAL FUNCTIONS

A removable discontinuity occurs in the graph of a rational function at $x=a$ if $a$ is a zero for a factor in the denominator that is common with a factor in the numerator. We factor the numerator and denominator and check for common factors. If we find any, we set the common factor equal to $0$ and solve. This is the location of the removable discontinuity. This is true if the multiplicity of this factor is greater than or equal to that in the denominator. If the multiplicity of this factor is greater in the denominator, then there is still an asymptote at that value.

EXAMPLE 6

Identifying Vertical Asymptotes and Removable Discontinuities for a Graph

Find the vertical asymptotes and removable discontinuities of the graph of $k(x)=\frac{x-2}{x^{2}-4}$.

Solution

Factor the numerator and the denominator.

$k(x)=\frac{x-2}{(x-2)(x+2)}$

Notice that there is a common factor in the numerator and the denominator, $x–2$. The zero for this factor is $x=2$. This is the location of the removable discontinuity.

Notice that there is a factor in the denominator that is not in the numerator, $x+2$. The zero for this factor is $x=−2$. The vertical asymptote is $x=−2$. See Figure 11.

Figure 11

The graph of this function will have the vertical asymptote at $x=−2$, but at $x=2$ the graph will have a hole.

TRY IT #5

Find the vertical asymptotes and removable discontinuities of the graph of $f(x)=\frac{x^{2}-25}{x^{3}-6 x^{2}+5 x}$.