Squeeze Theorem Visualization
About This MicroSim
The Squeeze Theorem (also called the Sandwich or Pinching Theorem) is a powerful tool for finding limits of functions that are difficult to evaluate directly but are bounded by simpler functions.
The Theorem
If \(g(x) \leq f(x) \leq h(x)\) for all x near c (except possibly at c), and:
\[\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L\]
Then:
\[\lim_{x \to c} f(x) = L\]
Examples Included
- x²sin(1/x): Squeezed between -x² and x² → limit is 0
- xcos(1/x): Squeezed between -|x| and |x| → limit is 0
- sin(x)/x: Squeezed between cos(x) and 1 → limit is 1
How to Use
- Example Selector: Choose different squeeze scenarios
- Distance Slider: Control how close to the target x-value
- Squeeze Button: Watch the squeeze zone narrow automatically
- Info Panel: See the bounds and limit value
Visual Elements
- Red curve: Upper bound h(x)
- Green curve: Middle function f(x) being squeezed
- Blue curve: Lower bound g(x)
- Shaded region: The "squeeze zone" where f(x) is trapped
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- State the Squeeze Theorem and its conditions
- Identify appropriate bounding functions for oscillating limits
- Apply the Squeeze Theorem to evaluate limits like \(\lim_{x \to 0} x^2 \sin(1/x)\)
Key Insight
The Squeeze Theorem works because if a function is trapped between two bounds, and both bounds converge to the same value, there's nowhere else for the middle function to go!
Suggested Activities
- Watch the Squeeze: Click "Squeeze" and observe how the shaded region narrows
- Compare Examples: Note which functions use the squeeze for bounded oscillation vs. other purposes
- Construct Bounds: For the function \(x \cdot \sin(x)\), what bounds would you use?
Assessment Questions
- Why does \(\lim_{x \to 0} x^2 \sin(1/x) = 0\) even though \(\sin(1/x)\) oscillates infinitely?
- What must be true about \(\lim g(x)\) and \(\lim h(x)\) for the Squeeze Theorem to apply?
- Could the Squeeze Theorem help find \(\lim_{x \to 0} \frac{\sin x}{x}\)?
Embedding
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