Limit of \( \displaystyle \frac{\sin{\left(6 x - 3 \right)}}{2 x - 1} \) as \( x \to \frac{1}{2} \)
Problem 1.524 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\sin{\left(6 x - 3 \right)}}{2 x - 1} \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\sin{\left(6 x - 3 \right)}}{2 x - 1}\right) \]limit factor algebraIdentify the limit to be evaluated. Factor out 3 from the argument of the sine function. This step is not needed, let's use substitution instead.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(3 u \right)}}{u}\right) \]substitutionLet u = 2*x - 1. As x approaches 1/2, u approaches 0.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \sin{\left(3 u \right)}}{\frac{d}{d u} u}\right) \]lhopitalThe limit is in the indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
- \[ = \lim_{u \to 0^+}\left(3 \cos{\left(3 u \right)}\right) \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = 3 \]limit simplifyEvaluate the limit by substituting u = 0. Final result.✓ Proved
Answer \( 3 \)
✓ 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 undefined where 2*x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where u = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 undefined where Derivative(u, u) = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(u, u) = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ 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: passqwen3.6:27b-mlx: fail (error) — Step 3 is mathematically nonsensical (multiplying by 1 in a convoluted way) and the note admits it is not needed, yet it is included as a formal step. Step 4 performs the substitution u = 2x - 1, but the transition from Step 3 to Step 4 is not a single valid algebraic or substitution step; it implicitly discards the garbage from Step 3 and changes the variable, violating the 'one thing per step' rule. Furthermore, Step 5 applies L'Hopital's rule, but the notation `Limit(Derivative(...)/Derivative(...), ...)` is not the standard application of the rule which usually results in a new limit of the ratio of derivatives, not a limit of the symbolic derivative objects themselves, though this is a minor notational issue compared to the broken flow in steps 3-4.
Senior review claude-sonnet-5-5, 2026-10-11: pass — Every line is equivalent to the previous one and the answer 3 is correct. Step 3 is a redundant multiply-by-1 whose note says it is not needed, which is sloppy but not wrong. Step 6 is labeled 'simplify' while it computes derivatives, which is acceptable as part of the L'Hopital step.
qwen3.6:27b-mlx: dismiss — Step 3 multiplies by (1/(2x-1))/(1/(2x-1)) = 1, which is a valid identity that SymPy proved, and step 4 follows from it by substitution, so there is no mathematical error. The note is awkward and the step is redundant, but that is a style issue.
Every verdict on record (5)
gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 3 is mathematically nonsensical (multiplying by 1 in a convoluted way) and the note admits it is not needed, yet it is included as a formal step. Step 4 performs the substitution u = 2x - 1, but the transition from Step 3 to Step 4 is not a single valid algebraic or substitution step; it implicitly discards the garbage from Step 3 and changes the variable, violating the 'one thing per step' rule. Furthermore, Step 5 applies L'Hopital's rule, but the notation `Limit(Derivative(...)/Derivative(...), ...)` is not the standard application of the rule which usually results in a new limit of the ratio of derivatives, not a limit of the symbolic derivative objects themselves, though this is a minor notational issue compared to the broken flow in steps 3-4.claude-sonnet-5-5: pass 2026-10-11 — Every line is equivalent to the previous one and the answer 3 is correct. Step 3 is a redundant multiply-by-1 whose note says it is not needed, which is sloppy but not wrong. Step 6 is labeled 'simplify' while it computes derivatives, which is acceptable as part of the L'Hopital step.qwen3.6:27b-mlx: fail (error) 2026-10-11 — Step 3 is mathematically nonsensical (multiplying by 1 in a convoluted way) and its note admits it is not needed, yet it is presented as a step in the solution. Step 4 performs a substitution that skips the invalid Step 3, making the sequence of steps logically disjointed and incorrect.gpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.