Home›Calculus 1›Trigonometric limits›Problem 1.377

Limit of \( \displaystyle \frac{\sin{\left(9 x - 3 \right)}}{3 x - 1} \) as \( x \to \frac{1}{3} \)

Problem 1.377 · medium

Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sin{\left(9 x - 3 \right)}}{3 x - 1} \).

\[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sin{\left(9 x - 3 \right)}}{3 x - 1}\right) \]
limit factorEvaluate the limit of the given function. Factor out 3 from the argument of the sine function.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(3 u \right)}}{u}\right) \]
substitutionLet u = 3*x - 1. As x approaches 1/3, u approaches 0.✓ Proved
\[ = \lim_{u \to 0^+}\left(\frac{3 \sin{\left(u \right)}}{u}\right) \]
algebraMove the constant 3 outside the limit.✓ Proved
\[ = 3 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right) \]
limit-lawApply the constant multiple rule for limits.✓ Proved
\[ = 3 \]
trig-limit simplifyUse the fundamental trigonometric limit limit_{u->0} sin(u)/u = 1. Final result.✓ Proved

Answer \( 3 \)

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 undefined where 3*x - 1 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 undefined where u = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where u = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 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
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 4 incorrectly rewrites sin(3u)/u as 3*sin(u)/u, which is not an algebraic simplification. The identity sin(3u)=3sin(u)-4sin^3(u) does not allow pulling a factor of 3 outside the fraction. This step applies more than one rule and introduces a false equality.
qwen3.6:27b-mlx: pass — The solution correctly factors the argument, performs a valid substitution, and applies standard limit laws and trigonometric limits. Each step changes only one aspect of the expression and uses an appropriate label from the vocabulary.

Senior review claude-sonnet-5-5, 2026-10-07: fail — Step 3 to 4 is invalid: sin(3u)/u is not 3*sin(u)/u, and SymPy's check only passed because both limits evaluate to 3. A valid route would rewrite sin(3u)/u as 3*sin(3u)/(3u) and substitute v=3u, or use a different substitution from the start.

gpt-oss:20b: uphold — Line 4 replaces sin(3u) with 3 sin(u) inside the limit, which is false; the two limits both equal 3 only by coincidence, and the note 'move the constant outside' does not justify it.

Every verdict on record (5)

gpt-oss:20b: fail (error) 2026-10-07 — Step 4 incorrectly rewrites sin(3u)/u as 3*sin(u)/u, which is not an algebraic simplification. The identity sin(3u)=3sin(u)-4sin^3(u) does not allow pulling a factor of 3 outside the fraction. This step applies more than one rule and introduces a false equality.
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly factors the argument, performs a valid substitution, and applies standard limit laws and trigonometric limits. Each step changes only one aspect of the expression and uses an appropriate label from the vocabulary.
claude-sonnet-5-5: fail (error) 2026-10-07 — Step 3 to 4 is invalid: sin(3u)/u is not 3*sin(u)/u, and SymPy's check only passed because both limits evaluate to 3. A valid route would rewrite sin(3u)/u as 3*sin(3u)/(3u) and substitute v=3u, or use a different substitution from the start.
qwen3.6:27b-mlx: pass 2026-10-07
gpt-oss:20b: fail (error) 2026-10-07 — Step 4 incorrectly rewrites sin(3*u) as 3*sin(u). The factor 3 cannot be pulled out of the sine function; sin(3u) is not equal to 3*sin(u).

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.