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Exercises Polynomials and factoring

Example 6 from Algebra and Trigonometry, 5.6 Rational Functions

Identifying Vertical Asymptotes and Removable Discontinuities for a Graph

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

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
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.

Your turn

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

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

How did it go?

PreviousIdentifying Vertical Asymptotes Next Identifying Horizontal and Slant 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.