Home›Calculus 1›L'Hôpital's rule›Problem 1.464

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

Problem 1.464 · easy

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

\[ \lim_{x \to \infty}\left(\frac{\ln{\left(x + 1 \right)}}{x + 1}\right) \]
limitStart with the original limit expression.✓ 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 in the indeterminate form infinity/infinity, so we apply L'Hôpital's rule.✓ Proved
\[ = \lim_{x \to \infty} \frac{1}{x} \]
simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the resulting fraction.✓ Proved
\[ = 0 \]
limitAs x approaches infinity, 1/x approaches 0.✓ Proved

Answer \( 0 \)

✓ 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. Thus the expression is altered by more than one rule and the subsequent steps are based on an incorrect intermediate form.
qwen3.6:27b-mlx: pass

Senior review claude-sonnet-5-5, 2026-10-09: pass — Every line is correct and carries an appropriate rule label. The substitution step renames the dummy variable u back to x, which is valid for limits. The L'Hôpital step and the subsequent simplifications are also valid.

gpt-oss:20b: dismiss — After substituting u = x+1, the limit variable is a dummy, so Limit(log(u)/u, u, oo) equals Limit(log(x)/x, x, oo); renaming the variable is legitimate and SymPy confirmed the equality.

Every verdict on record (5)

gpt-oss:20b: fail (error) 2026-10-09 — 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. Thus the expression is altered by more than one rule and the subsequent steps are based on an incorrect intermediate form.
qwen3.6:27b-mlx: pass 2026-10-09
claude-sonnet-5-5: pass 2026-10-09 — Every line is correct and carries an appropriate rule label. The substitution step renames the dummy variable u back to x, which is valid for limits. The L'Hôpital step and the subsequent simplifications are also valid.
gpt-oss:20b: fail (error) 2026-10-09 — Step 2 incorrectly replaces log(x+1)/(x+1) with log(x)/x. The substitution u=x+1 would give log(u)/u, not log(x)/x. This changes the expression and invalidates the subsequent steps.
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies substitution to simplify the expression, followed by L'Hôpital's rule and standard limit evaluation. Each step adheres to the one-change-per-step constraint and uses valid labels from the vocabulary.

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 2026-10-09 with SymPy 1.14.0.