Sin(x)/x Visualization
About This MicroSim
This is arguably the most important limit in calculus:
\[\lim_{x \to 0} \frac{\sin x}{x} = 1\]
where x is measured in radians. This limit is fundamental to differentiating sine and cosine functions.
Dual Visualization
- Left side (Unit Circle): Shows the geometric relationship between arc length x and sin(x)
- Right side (Graph): Shows the function y = sin(x)/x with a hole at x = 0
The Key Insight
For small angles, the arc length (x) and the vertical height (sin x) become nearly equal. As x approaches 0, their ratio approaches 1.
How to Use
- X Slider: Adjust the angle in radians
- Show Table Checkbox: Display numerical values showing convergence
- Observe: Watch how the ratio sin(x)/x approaches 1 as x → 0
Visual Elements
- Blue arc: The arc of length |x| on the unit circle
- Red line: The vertical height sin(x)
- Green curve: The function sin(x)/x
- Magenta point: Current x value on the graph
Lesson Plan
Learning Objectives
After using this MicroSim, students will be able to:
- Explain geometrically why \(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
- Understand why this limit requires x in radians
- Apply this limit to evaluate related trigonometric limits
Key Concept
On the unit circle, x (in radians) represents arc length. As the angle shrinks, the arc becomes nearly vertical, making arc length ≈ sin(x).
Suggested Activities
- Geometric Exploration: On the unit circle view, compare arc and sin visually as x decreases
- Numerical Evidence: Enable the table to see the ratio converging to 1
- Graph Analysis: Identify the hole at x = 0 and the horizontal asymptote-like behavior
Assessment Questions
- What is \(\lim_{x \to 0} \frac{\sin(3x)}{x}\)? (Hint: Rewrite as \(3 \cdot \frac{\sin(3x)}{3x}\))
- Why doesn't this limit work if x is in degrees?
- Use the fundamental limit to find \(\lim_{x \to 0} \frac{\sin x}{2x}\)
Related Limit
A companion limit you should also know:
\[\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\]
Embedding
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