CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The limit of sin x divided by x at zero is one

Statement

$\lim_{x \to 0} \frac{\sin x}{x} = 1.$

Facts & Assumptions

Given: The sine function at zero.

[L1]

$\sin^{'} (0) = \cos 0 = 1$ and $\sin 0 = 0$ (The derivatives of sine and cosine are cosine and minus sine).

[L2]

The derivative at $0$ is $\lim_{x \to 0} (\sin x - \sin 0) / (x - 0)$ (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set).

Proof

technique · direct

1.1

Substituting [L1] into the derivative definition [L2] gives $\lim_{x \to 0} \sin x / x = 1$ .

L1L2

2.1

This is the asserted limit.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

13 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

NIST Digital Library of Mathematical Functions, Chapter 4 (standard reference, not scraped)
C. Schmeiser, Introduction to Analysis (standard reference, not scraped)