Limit of \( \displaystyle \frac{\sin{\left(8 x + 4 \right)}}{6 x + 3} \) as \( x \to - \frac{1}{2} \)
Problem 1.145 · medium
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\sin{\left(8 x + 4 \right)}}{3 \left(2 x + 1\right)} \).
- \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\sin{\left(8 x + 4 \right)}}{6 x + 3}\right) \]limit algebraSet up the limit. Rewrite the numerator using factoring.✓ Proved
- \[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \sin{\left(8 x + 4 \right)}}{\frac{d}{d x} \left(6 x + 3\right)}\right) \]lhopitalApply L'Hôpital's Rule because we have a 0/0 indeterminate form.✓ Proved
- \[ = \lim_{x \to - \frac{1}{2}^+}\left(\frac{4 \cos{\left(8 x + 4 \right)}}{3}\right) \]lhopital simplifyCompute the derivatives of the numerator and denominator. Simplify the constant fraction.✓ Proved
- \[ = \frac{4}{3} \]limit simplify simplify simplifyEvaluate the limit by substituting x = -1/2. Simplify the argument of the cosine function. Evaluate the cosine of zero. Final value.✓ Proved
Answer \( \frac{4}{3} \)
✓ 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 6*x + 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6*x + 3 = 0 undefined where Derivative(6*x + 3, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(6*x + 3, x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 9 | ✓ 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 (style) — Step 4 is labeled 'lhopital' but performs the differentiation and simplification of the resulting fraction, violating the one-rule-per-step constraint. It should be split into a 'derivative' step and a 'simplify' step.
Senior review claude-sonnet-5-5, 2026-10-04: pass — All lines are SymPy-verified. Each step names a legitimate rule from the vocabulary. The 0/0 form at x=-1/2 justifies L'Hôpital, and the result 4/3 is correct.
qwen3.6:27b-mlx: dismiss — 'derivative' is not in the allowed vocabulary, so 'lhopital' is a legitimate label for evaluating the derivatives within the L'Hôpital step; and for step 2, 'algebra' is acceptable when 'factor' also fits (8x+4 = 4(2x+1)).
Every verdict on record (5)
qwen3.6:27b-mlx: fail (style) 2026-10-04 — Step 4 is labeled 'lhopital' but performs the differentiation and simplification of the resulting fraction, violating the one-rule-per-step constraint. It should be split into a 'derivative' step and a 'simplify' step.gpt-oss:20b: pass 2026-10-04claude-sonnet-5-5: pass 2026-10-04 — All lines are SymPy-verified. Each step names a legitimate rule from the vocabulary. The 0/0 form at x=-1/2 justifies L'Hôpital, and the result 4/3 is correct.qwen3.6:27b-mlx: fail (error) 2026-10-04 — Step 4 is labeled 'lhopital', but it performs differentiation, which is a computational step following the application of L'Hôpital's Rule (done in Step 3). Differentiation should be labeled 'derivative'. Additionally, Step 2 is labeled 'algebra' but performs factoring, so 'factor' is the more precise label from the vocabulary, though 'algebra' is acceptable if 'factor' is not strictly required for correctness; however, the mislabeling of the differentiation step is a clear defect.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.