Limit Visualization with Hole
About This MicroSim
This visualization demonstrates the core concept of limits: what value a function approaches as x gets closer to a target, even when the function is undefined at that exact point.
The function shown is:
\[f(x) = \frac{x^2 - 4}{x - 2}\]
This simplifies to \(f(x) = x + 2\) for all \(x \neq 2\), but has a hole at \(x = 2\) where the original expression is undefined (0/0).
Key Observations
- As x approaches 2 from either side, f(x) approaches 4
- The function value at x = 2 doesn't exist (there's a hole)
- The limit still equals 4 because we care about what the function approaches, not what it equals
How to Use
- Distance Slider: Adjust how close x is to 2
- Approach Direction: Choose to approach from left, right, or both simultaneously
- Animate: Watch the approach happen automatically
- Reset: Return to starting position
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- Explain that a limit describes approach behavior, not the actual function value
- Interpret limit notation \(\lim_{x \to c} f(x) = L\)
- Identify when a function has a removable discontinuity (hole)
Suggested Activities
- Predict and Observe: Before animating, have students predict what y-value the function approaches
- Compare Approaches: Switch between left, right, and both approaches to verify they give the same limit
- Discussion: Why does the limit exist even though f(2) is undefined?
Assessment Questions
- What is \(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)?
- Is f(2) defined? Why or why not?
- How is the limit different from the function value?
Embedding
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