Lesson 6: 1-6 Algebraic Manipulation
Duration of Days: 1
Lesson Objective
The student will be able to:
Identify the indeterminate form 0/0 as a signal that more algebraic work is required.
Apply the Factoring and Canceling technique to remove removable discontinuities.
Utilize Rationalization (using the conjugate) to evaluate limits involving square roots.
Simplify Complex Fractions to find limits of nested rational expressions.
Understand that the resulting simplified function is identical to the original except at the point of the hole.
Why does getting 0/0 not mean the limit "Does Not Exist"?
How does "canceling a factor" change the graph of a function without changing its limit?
When is it appropriate to use the conjugate of an expression?
Indeterminate Form
Removable Discontinuity
Factor and Cancel
Conjugate
Rationalization
Complex Fraction
Domain Restriction
MA.F-IF.A.2
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Description: This lesson teaches students how to "repair" a function to find its limit. When direct substitution yields 0/0, it usually indicates a common factor in the numerator and denominator that is "hiding" the limit.
Purpose: This is a vital algebraic skill for the AP Exam. It reinforces the idea that the limit cares about the neighborhood of a point, not the point itself. Canceling the "problem factor" allows us to see where the graph was heading before it hit the "pothole."
DOK Level 2 (Skill/Concept): Factoring a basic quadratic to evaluate a limit.
DOK Level 3 (Strategic Thinking): Choosing the correct algebraic strategy (Rationalizing vs. Complex Fractions) based on the structure of a given function.
For Struggling Learners (Scaffolding):
The "Indeterminate" Warning: Tell students that 0/0 is like a "Keep Digging" sign. It doesn't mean "stop" or "undefined"—it means "there is an answer, but it's hidden."
For Advanced Learners (Extension):Difference of Cubes: Provide limits that require the sum or difference of cubes formula.
Creating Functions: Challenge students to write a function that has a limit o but results in 0/0 upon direct substitution.
AP College Board AP Assessments