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Exercises Polynomials and factoring

Example 7 from Algebra and Trigonometry, 5.6 Rational Functions

Identifying Horizontal and Slant Asymptotes

For the functions listed, identify the horizontal or slant asymptote.

ⓐ $g (x) = \frac{6 x^{3} - 10 x}{2 x^{3} + 5 x^{2}}$
ⓑ $h (x) = \frac{x^{2} - 4 x + 1}{x + 2}$
ⓒ $k (x) = \frac{x^{2} + 4 x}{x^{3} - 8}$

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.

1
Solution
For these solutions, we will use $f (x) = \frac{p (x)}{q (x)}, q (x) \neq 0.$
1. ⓐ $g (x) = \frac{6 x^{3} - 10 x}{2 x^{3} + 5 x^{2}} :$ The degree of $p = degree of q = 3,$ so we can find the horizontal asymptote by taking the ratio of the leading terms. There is a horizontal asymptote at $y = \frac{6}{2}$ or $y = 3.$
2. ⓑ $h (x) = \frac{x^{2} - 4 x + 1}{x + 2} :$ The degree of $p = 2$ and degree of $q = 1.$ Since $p = 2 <$ by 1, there is a slant asymptote found at $\frac{x^{2} - 4 x + 1}{x + 2} .$
  The quotient is $x - 6$ and the remainder is 13. There is a slant asymptote at $y = x - 6.$
3. ⓒ $k (x) = \frac{x^{2} + 4 x}{x^{3} - 8} :$ The degree of $p = 2 <$ degree of $q = 3,$ so there is a horizontal asymptote $y = 0.$

How did it go?

PreviousIdentifying Vertical Asymptotes and Removable Discontinuities for a Graph Next Identifying Horizontal Asymptotes

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.