Limit of \( \displaystyle \frac{\ln{\left(x - 1 \right)}}{x - 1} \) as \( x \to \infty \)
Problem 1.164 · easy
- \[ \lim_{x \to \infty}\left(\frac{\ln{\left(x - 1 \right)}}{x - 1}\right) \]limitIdentify the limit to be evaluated.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\ln{\left(x \right)}}{x}\right) \]substitutionLet u = x - 1, then as x approaches infinity, u also approaches infinity.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\frac{d}{d x} \ln{\left(x \right)}}{\frac{d}{d x} x}\right) \]lhopitalThe limit is of the indeterminate form infinity/infinity, so apply L'Hopital's rule.✓ Proved
- \[ = \lim_{x \to \infty} \frac{1}{x} \]simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the resulting expression.✓ Proved
- \[ = 0 \]limitEvaluate the limit as x approaches infinity.✓ Proved
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | line 1 is the problem as stated |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x - 1 = 0 undefined where x = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where x = 0 undefined where Derivative(x, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where Derivative(x, x) = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| answer | ✓ Proved | sympy 1.14.0 | final line against the stated answer: simplify(a - b) reduced to 0 |
| answer, a second way | ✓ Proved | sympy 1.14.0 | SymPy took the limit from both sides and got the stated value |
Reviewers
gpt-oss:20b: fail (error) — Step 2 incorrectly replaces “log(x-1)/(x-1)” with “log(x)/x”. The substitution u=x-1 yields “log(u)/u”, not “log(x)/x”. This changes the expression and invalidates the subsequent steps.qwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-04: pass — The solution is correct: the substitution (with the variable renamed), L'Hopital, simplification and evaluation are all valid, and the answer 0 is right. The step labels fit the vocabulary.
gpt-oss:20b: dismiss — Substituting u = x - 1 gives log(u)/u, and renaming the dummy variable u back to x gives Limit(log(x)/x, x, oo), which is the same limit. SymPy confirmed the two lines are equal.
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (error) 2026-10-04 — Step 2 incorrectly replaces “log(x-1)/(x-1)” with “log(x)/x”. The substitution u=x-1 yields “log(u)/u”, not “log(x)/x”. This changes the expression and invalidates the subsequent steps.claude-sonnet-5-5: pass 2026-10-04 — The solution is correct: the substitution (with the variable renamed), L'Hopital, simplification and evaluation are all valid, and the answer 0 is right. The step labels fit the vocabulary.qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (error) 2026-10-04 — Step 2 incorrectly replaces “log(x-1)/(x-1)” with “log(x)/x”. The substitution u=x-1 yields log(u)/u, not log(x)/x, so the subsequent steps are based on an invalid expression.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
gemma4:26b, checked 2026-10-04 with SymPy 1.14.0.