Lesson 8: 1-8 The Squeeze Theorem
Duration of Days: 1
Lesson Objective
The student will be able to:
Apply the Squeeze Theorem (also known as the Sandwich Theorem) to find the limit of a function trapped between two other functions.
Manipulate trigonometric expressions (using substitution or properties) to transform them into a form where the special limits can be applied.
If function g is always between functions f and h, and f and h both approach the same limit at a point, what must happen to g?
How does the behavior of sin(x) over x differ from the standard special limit?
Squeeze Theorem / Sandwich Theorem
Trigonometric Limit
Oscillating Behavior
Bounded Function
Argument of a Function
"Pinned" Function
MA.F-IF.B.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.
Description: The Squeeze Theorem is a logical tool. This is often used for functions involving oscillation. Additionally, students learn to recognize functions.
Purpose: This lesson introduces students to mathematical rigor beyond basic algebra. It teaches them how to handle "non-algebraic" functions (transcendental functions) and provides a conceptual bridge to proving derivatives.6.
DOK Levels
DOK Level 2 (Skill/Concept): Evaluating a limit by adjusting the coefficients.
DOK Level 3 (Strategic Thinking): Constructing a Squeeze Theorem argument by identifying two "bounding" functions.
For Struggling Learners (Scaffolding):
The "Two Guards" Analogy: Explain that the top and bottom functions are like two guards walking a prisoner (the middle function) to a specific jail cell (the limit).
For Advanced Learners (Extension):Analytical Proofs: Challenge students to use the unit circle and area comparisons to prove why is bounded by cos x and 1.
AP College Board Assessments