Pre-Calculus 12 Notes

By: Gen L.

In partnership with Hyperion University, 2024

Lesson 4: Points of Discontinuity

Levels of Continuity

Throughout Math, you will find many levels of continuity.

  1. Absolute Continuity (Limit and Value Continuous)
  2. Removable Discontinuity or "hole" (Limit Continuous)
  3. Asymptotic Discontinuity ( as )
  4. Absolute Discontinuity or "gap" (Limit and Value Discontinuous)

Last time, we focused on Asymptotic Discontinuity.
Now, let's look at Removable / Point Discontinuities.

Points of Discontinuity

  • PODs are singular points missing or separate from the graph.
  • They result from simplification, often cancelling something out.
  • They are drawn as holes in the graph.

Examples

Try graphing . We are left with .

Since it's a line, there is no asymptotes.

The behaviour at : . But, . It's a hole.

As well, try graphing .

Characteristics to Equation

Write the equation of a possible rational function with the following characteristics:

  • A vertical asymptote at
  • A removable discontinuity at
  • x-intercept of -3

Determining Horizontal Asymptotes

  • Look at the dominant terms (Degree).
    • ,
    • , no HA.
    • ,

Determine Characteristics

  • -int: set numerator =
  • -int: set
  • Restrictions: set denominator =
  • VA: Factor not in numerator
  • POD: Factor also in numerator
  • Domain: use restrictions
  • Range: use HA and PODs

Determine Characteristics, cont.

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