Lesson 13: 1-13 Removing Discontinuities
Duration of Days: 1
Lesson Objective
The student will be able to:
Identify removable discontinuities (holes) using algebraic methods (factoring and simplifying).
Redefine a function at a single point to create a "continuous extension."
Construct a piecewise function that "plugs the hole" of a rational function.
Distinguish between a hole that can be removed and a vertical asymptote that cannot.
Why is a hole called a "removable" discontinuity, but a vertical asymptote is "non-removable"?
If we cancel a factor like (x-2) from the numerator and denominator, what information about the original function are we "losing"?
Removable Discontinuity
Continuous Extension
Point Discontinuity
Singular Point
Redefinition
Non-Removable Discontinuity
F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
F-IF.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
F-IF.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
F-BF.1: Write a function that describes a relationship between two quantities.
Description: This lesson is the surgical side of limits. Students take a function which has a hole at x=3, and "repair" it. They learn that by simplifying the function and finding the limit, they can define a new piecewise function.
Purpose: This lesson reinforces the relationship between limits and function values. It teaches students that the limit tells us exactly what the "missing" value should be. This concept is vital for understanding L'Hôpital's Rule and certain proofs in higher-level analysis.
DOK Level 2 (Skill/Concept): Finding the coordinates (c, L) of a hole and writing the simplified version of the function.
DOK Level 3 (Strategic Thinking): Creating a continuous extension for a function that involves more complex algebra, such as rationalizing a numerator.
For Struggling Learners (Scaffolding):
The "Pothole Repair" Analogy: The limit is the material used to fill the pothole. If you don't know the limit, you don't know how much "fill" you need to make the road smooth.
Advanced Learners:
Multiple Holes: Provide a function with two holes and one asymptote, then ask them to write the piecewise function that removes all removable discontinuities.
AP College Board Assessments