Vertical and Horizontal Asymptotes of Rational Functions

This section will dive deeper into the analytic and algebraic tools for finding the three types of asymptotes found in rational functions. Defining asymptotes will help you graph rational functions without a calculator, determine where the function is undefined, and give you a picture of the general behavior of the function.

Identifying Vertical Asymptotes of Rational Functions

Vertical Asymptotes

The vertical asymptotes of a rational function may be found by examining the factors of the denominator that are not common to the factors in the numerator. Vertical asymptotes occur at the zeros of such factors.

HOW TO

Given a rational function, identify any vertical asymptotes of its graph.

1. Factor the numerator and denominator.

2. Note any restrictions in the domain of the function.

3. Reduce the expression by canceling common factors in the numerator and the denominator.

4. Note any values that cause the denominator to be zero in this simplified version. These are where the vertical asymptotes occur.

5. Note any restrictions in the domain where asymptotes do not occur. These are removable discontinuities, or "holes".

EXAMPLE 5

Identifying Vertical Asymptotes

Find the vertical asymptotes of the graph of $f(x)=\frac{5+2 x^{2}}{2-x-x^{2}}$ .

Solution

First, factor the numerator and denominator.

$\begin{aligned} k(x) &=\frac{5+2 x^{2}}{2-x-x^{2}} \\ &=\frac{5+2 x^{2}}{(2+x)(1-x)} \end{aligned}$

To find the vertical asymptotes, we determine where this function will be undefined by setting the denominator equal to zero:

$\begin{aligned} (2+x)(1-x) &=0 \\ x &=-2,1 \end{aligned}$

Neither $x=–2$ nor $x=1$ are zeros of the numerator, so the two values indicate two vertical asymptotes. The graph in Figure 9 confirms the location of the two vertical asymptotes.

Figure 9