Home›Calculus 1›L'Hôpital's rule›Problem 1.151

Limit of \( \displaystyle \frac{\ln{\left(2 x - 1 \right)}}{2 x - 1} \) as \( x \to \infty \)

Problem 1.151 · medium

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{\ln{\left(2 x - 1 \right)}}{2 x - 1} \).

\[ \lim_{x \to \infty}\left(\frac{\ln{\left(2 x - 1 \right)}}{2 x - 1}\right) \]
limitStart with the original limit.✓ Proved
\[ = 0 \]
substitutionSubstitute u = 2*x - 1.✓ Proved
\[ = \lim_{u \to \infty}\left(\frac{\ln{\left(u \right)}}{u}\right) \]
limitEvaluate the limit 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
\[ = 0 \]
limitThe limit is 0.✓ 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 2*x - 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies substitution, L'Hopital's rule, and limit evaluation. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-04: fail — The final answer 0 is correct and the L'Hopital chain in lines 3-7 is fine. Line 2 is a malformed conditional expression rather than the substituted limit Limit(log(u)/u, u, oo), so the substitution step is not validly stated and its SymPy check proves nothing.

qwen3.6:27b-mlx: uphold — Line 2 is a Python conditional (`... if u == 2*x - 1 else 0`), not a mathematical expression equal to line 1. Its 'proved' status is vacuous, since the conditional evaluates to 0 and so the SymPy check against the neighbouring lines was meaningless. The claim that step 3 violates the single-rule constraint is overstated, but the core objection about step 2 holds.

Every verdict on record (5)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies substitution, L'Hopital's rule, and limit evaluation. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: pass 2026-10-04
claude-sonnet-5-5: fail (error) 2026-10-04 — The final answer 0 is correct and the L'Hopital chain in lines 3-7 is fine. Line 2 is a malformed conditional expression rather than the substituted limit Limit(log(u)/u, u, oo), so the substitution step is not validly stated and its SymPy check proves nothing.
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 2 is syntactically invalid and does not represent a valid algebraic substitution step; it introduces a conditional expression rather than transforming the limit expression. Step 3 then arbitrarily discards the conditional structure without justification, violating the single-rule-per-step constraint and logical continuity.
gpt-oss:20b: pass 2026-10-04

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.