False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

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A sector-area squeeze proves lim sin(x)/x=1 without first calibrating angle measure

Statement

False claim: the sector-area inequalities $\sin θ < θ < \tan θ$ prove $\lim_{x \to 0} \sin x / x = 1$ without any prior calibration of the angle variable $θ$ to arc length, sector area, and $π$ .

Facts & Assumptions

Given: The analytic sine function and the definition $π = 2 γ$ from the first positive cosine zero.

[L1]

The analytic proof already establishes $\lim_{x \to 0} \sin x / x = 1$ (The limit of sin x divided by x at zero is one).

[L2]

$π$ is defined analytically from cosine, not from geometric sector area (Pi as twice the smallest positive zero of cosine).

Refutation

technique · direct

1.1

The sector inequality uses an angle measured in radians, and its usual derivation identifies that measure through arc length or sector area in the unit circle.

given

1.2

Establishing that identification requires a normalization constant, equivalently the relationship between the geometric full turn and the analytic $π$ of [L2].

2.1

Thus the sector argument cannot be used as a foundation independent of that calibration; it may prove the limit only after importing the relation whose analytic construction it was meant to justify.

step 1.1step 1.2

3.1

The limit itself remains true by [L1], but the claimed independent proof is false.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. Schmeiser, Introduction to Analysis (standard reference, not scraped)
R. Bartle and D. Sherbert, Introduction to Real Analysis (standard reference, not scraped)
H. Zeisel, lim sin(x)/x and the definition of pi (standard reference, not scraped)