Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02
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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 logarithm grows more slowly than every positive real power

Statement

For every α>0, lim⁡x→+∞log⁡xxα=0.

Facts & Assumptions

Given: A real α>0.

[L1]

exp⁡(t) tends to +∞ as t→+∞, and its range is (0,∞) (The exponential tends to +∞ at +∞ and to 0 at −∞, The natural logarithm as the inverse of the exponential function).

[L2]

For a>0, t/exp⁡(at)→0 as t→+∞ (The exponential dominates every fixed nonnegative integer power at +∞).

[L3]

xα=exp⁡(αlog⁡x) for x>0, and algebra of limits permits substitution through the displayed identities (Real powers for positive bases, with the zero-base positive-exponent convention, Algebra of limits: sums, scalar multiples, products and quotients).

Proof

technique · direct
1.1L1

Put t=log⁡x. As x→+∞, the inverse relation and [L1] give t→+∞.

1.2L3

By [L3], log⁡x/xα=t/exp⁡(αt).

2.1step 1.1step 1.2L2∎

The right-hand side tends to 0 by [L2], which proves the claim.

Depends on

Dependency tree · 2 levels

Sources