Finding Removable Discontinuities

Description/Explanation/Highlights

Video Description

In this video we will learn how to find removable discontinuities in a rational function and distinguish them from vertical asymptotes.

Steps and Key Points to Remember

To find a removable discontinuity in a rational function, follow these steps:

  1. Sometimes when finding vertical asymptotes for rational functions, what appears to be a vertical asymptote is actually a removable discontinuity.
  2. Removable discontinuities are “holes” in a graph instead of an asymptote that the graph approaches but never reaches.
  3. To find removable discontinuities, first look for the possible vertical asymptotes by setting the denominator equal to 0 and solving for x.
  4. If you need help finding the vertical asymptotes, please see the video Finding Vertical Asymptotes.
  5. To find a removable discontinuity or to see if one exists, begin by factoring the numerator and denominator completely.
  6. Set each factor in the denominator = 0 and solve for x. These are the potential vertical asymptotes or removable discontinuities.
  7. If there is a common factor in both the numerator and denominator, the result from that factor is a removable discontinuity instead of a vertical asymptote.
  8. Fully factor the example \(f(x)=\frac{\textstyle x^2-4}{\textstyle x^2-7x+10}\) to \(f(x)=\frac{\textstyle (x+2)(x-2)}{\textstyle (x-5)(x-2)}\).
  9. Set both factors in the denominator equal to 0 and solve.  \(x-5=0\) and \(x-2=0\). So \(x=5\) and \(x=2\) are vertical asymptotes or removable discontinuities.
  10. Since there is a common factor of \(x-2\) in the numerator and denominator, the result \(x=2\) is a removable discontinuity instead of a vertical asymptote. \(x=5\) remains a vertical asymptote since there is no matching factor in the numerator.

Here are some key points to keep in mind when finding removable discontinuities.

  • Removable discontinuities are found in the same way vertical asymptotes are found but are distinguished from vertical asymptotes by factors in the numerator and denominator that are exactly the same.
  • A removable discontinuity appears on the graph as a “hole” in the graph.
  • A graph could have no vertical asymptotes and multiple removable discontinuities, both vertical asymptotes and removable discontinuities, or only vertical asymptotes depending on the factors.

Video Highlights

  • 00:00 Introduction
  • 00:23 \(y=\frac{\textstyle x^2-4}{\textstyle x^2-7x+10}\) example of finding removable discontinuities
  • 03:19 \(y=\frac{\textstyle x^2+7x+12}{\textstyle x^2+x-6}\) example resulting in both a removable discontinuity and a vertical asymptote
  • 04:45 \(y=\frac{\textstyle x^2-2x}{\textstyle 3x-6}\) example with no vertical asymptote
  • 05:32 \(y=\frac{\textstyle x^2-x-2}{\textstyle x^2-5x-4}\) example with no removable discontinuity
  • 07:10 Conclusion
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