Limit of \( \displaystyle \frac{\ln{\left(3 x + 1 \right)}}{3 x + 1} \) as \( x \to \infty \)
Problem 1.362 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\ln{\left(3 x + 1 \right)}}{3 x + 1} \).
- \[ \lim_{x \to \infty}\left(\frac{\ln{\left(3 x + 1 \right)}}{3 x + 1}\right) \]limitStart with the original limit.✓ Proved
- \[ = \lim_{u \to \infty}\left(\frac{\ln{\left(3 u + 1 \right)}}{3 u + 1}\right) \]substitutionSubstitute u for x.✓ Proved
- \[ = \lim_{u \to \infty}\left(\frac{\ln{\left(u \right)}}{u}\right) \]simplifySimplify the expression as u approaches infinity.✓ Proved
- \[ = \lim_{u \to \infty}\left(\frac{\frac{d}{d u} \ln{\left(u \right)}}{\frac{d}{d u} u}\right) \]lhopitalApply L'Hopital's rule.✓ Proved
- \[ = \lim_{u \to \infty} \frac{1}{u} \]simplify simplifyCompute the derivatives. Simplify the fraction.✓ Proved
- \[ = 0 \]limitEvaluate the final limit.✓ Proved
Answer \( 0 \)
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. A reviewer's objection to the explanation stands; see the receipt.
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 3*x + 1 = 0 undefined where 3*u + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where 3*u + 1 = 0 undefined where u = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where u = 0 undefined where Derivative(u, u) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where Derivative(u, u) = 0 undefined where u = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 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 3 incorrectly rewrites log(3*u+1)/(3*u+1) as log(u)/u, which changes the expression. The substitution in step 2 was unnecessary and the simplification in step 3 is invalid.qwen3.6:27b-mlx: fail (error) — Step 3 incorrectly simplifies log(3*u + 1)/(3*u + 1) to log(u)/u. While the limit is the same, the expressions are not algebraically equal, violating the constraint that consecutive lines must be algebraically equal. This is not a valid 'simplify' step.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-07 — Step 3 incorrectly rewrites log(3*u+1)/(3*u+1) as log(u)/u, which changes the expression. The substitution in step 2 was unnecessary and the simplification in step 3 is invalid.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 3 incorrectly simplifies log(3*u + 1)/(3*u + 1) to log(u)/u. While the limit is the same, the expressions are not algebraically equal, violating the constraint that consecutive lines must be algebraically equal. This is not a valid 'simplify' step.gpt-oss:20b: fail (error) 2026-10-07 — Step 3 incorrectly rewrites log(3*u+1)/(3*u+1) as log(u)/u, changing the expression. This applies more than one transformation and is not justified by any listed rule.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 3 incorrectly simplifies log(3*u + 1)/(3*u + 1) to log(u)/u. While the limit is the same, the expressions are not algebraically equal (e.g., at u=1, LHS=log(4)/4, RHS=0), violating the contract that consecutive lines must be algebraically equal. Step 3 should have been a limit-law or asymptotic argument, not a simplification.
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 2026-10-07 with SymPy 1.14.0.