The squeeze theorem: why sin x / x tends to 1
This animation visualizes why sin x / x approaches 1 as x approaches 0, even though the function itself is undefined there. It shows cos x and the constant function 1 forming a narrowing band around sin x / x near the origin, illustrating the Squeeze Theorem in action. As x shrinks toward 0, both bounding curves converge to 1, forcing the trapped function to the same limit. Useful for students learning limits and trigonometric identities in introductory calculus.
16:9 · every frame verified for overlaps, spacing and edges before rendering
The prompt that made it
sin x / x is undefined at 0 but trapped between cos x and 1 near 0; as the band narrows both bounds go to 1, so the limit is 1.
Make the next one in this series
- Show why the Squeeze Theorem requires both bounding functions to share the same limit at a point.
- Animate the derivative of sin x derived directly from this limit using the definition of a derivative.
- Compare this limit visually to (1 - cos x)/x approaching 0 near the origin.
