Home›Calculus 1›L'Hôpital's rule›Problem 1.796

Limit of \( \displaystyle \frac{\ln{\left(3 x + 1 \right)}}{3 x} \) as \( x \to 0 \)

Problem 1.796 · medium Video

Evaluate \( \displaystyle \lim_{x \to 0} \frac{\ln{\left(3 x + 1 \right)}}{3 x} \).

\[ \lim_{x \to 0^+}\left(\frac{\ln{\left(3 x + 1 \right)}}{3 x}\right) \]
limit algebraIdentify the limit to be evaluated. Rearrange the terms in the numerator.✓ Proved
\[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \ln{\left(3 x + 1 \right)}}{\frac{d}{d x} 3 x}\right) \]
lhopitalThe limit is an indeterminate form 0/0, so apply L'Hôpital's rule.✓ Proved
\[ = \lim_{x \to 0^+} \frac{1}{3 x + 1} \]
simplify simplifyCalculate the derivatives of the numerator and denominator. Simplify the resulting fraction.✓ Proved
\[ = 1 \]
limitEvaluate the limit by substituting x = 0.✓ Proved

Answer \( 1 \)

The graph of f(x) = log(3x + 1)/(3x). Two points slide along the curve toward x = 0, one from each side; their heights close in on 1.

✓ 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. How we verify

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 = 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(3*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(3x, x) = 0 undefined where 3x + 1 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x + 1 = 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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-01
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies L'Hôpital's rule and simplifies the resulting expression to find the limit. Each step changes only one aspect of the expression and uses valid labels from the provided vocabulary.
gpt-oss:20b: pass 2026-10-31
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly identifies the indeterminate form, applies L'Hôpital's rule, computes the derivatives, and simplifies to the correct limit. Each step changes only one aspect of the expression and uses valid labels.
gpt-oss:20b: pass 2026-10-31

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 2026-11-01 with SymPy 1.14.0.