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Exercises Polynomials and factoring

Example 9 from Algebra and Trigonometry, 5.6 Rational Functions

Identifying Horizontal and Vertical Asymptotes

Find the horizontal and vertical asymptotes of the function

f (x) = \frac{(x - 2) (x + 3)}{(x - 1) (x + 2) (x - 5)}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
First, note that this function has no common factors, so there are no potential removable discontinuities.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The denominator will be zero at $x = 1, - 2, and 5,$ indicating vertical asymptotes at these values.
The numerator has degree 2, while the denominator has degree 3. Since the degree of the denominator is greater than the degree of the numerator, the denominator will grow faster than the numerator, causing the outputs to tend towards zero as the inputs get large, and so as $x \to \pm \infty, f (x) \to 0.$ This function will have a horizontal asymptote at $y = 0.$ See Figure 15.

Figure 15

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Find the vertical and horizontal asymptotes of the function:

$f (x) = \frac{(2 x - 1) (2 x + 1)}{(x - 2) (x + 3)}$

How did it go?

PreviousIdentifying Horizontal Asymptotes Next Finding the Intercepts of a Rational Function

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 5.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.