Limit of \( \displaystyle \frac{\sin{\left(8 x - 4 \right)}}{6 x - 3} \) as \( x \to \frac{1}{2} \)
Problem 1.46 · medium Video
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 algebraIdentify the limit to be evaluated. Factor out 4 from the argument of the sine function.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \sin{\left(8 x - 4 \right)}}{\frac{d}{d x} \left(6 x - 3\right)}\right) \]lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{4 \cos{\left(8 x - 4 \right)}}{3}\right) \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = \frac{4}{3} \]limit simplifyEvaluate the limit by substituting x = 1/2. Simplify the final numerical expression.✓ 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. 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 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 |
| 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 3 applies L’Hôpital’s rule but computes the derivatives incorrectly: the derivative of sin(4(2x‑1)) is 8 cos(8x‑4), not 4 cos(4(2x‑1)), and the derivative of 3(2x‑1) is 6, not 3. This makes the subsequent steps invalid.qwen3.6:27b-mlx: pass — The solution correctly identifies the indeterminate form and applies L'Hopital's rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Senior review claude-sonnet-5-5, 2026-10-03: pass — The solution is correct. The factoring, the 0/0 L'Hôpital step, the derivative evaluation (8cos/6 reduced to 4cos/3), the substitution and the final simplification are all valid, and the rule labels fit.
gpt-oss:20b: dismiss — Line 3 only writes the unevaluated Derivative objects, so no derivative is miscomputed there; line 4 evaluates them as 8cos(...)/6 = 4cos(...)/3, which is correct, and SymPy proved it.
Every verdict on record (7)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly identifies the indeterminate form and applies L'Hopital's rule. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-03 — Step 3 applies L’Hôpital’s rule but computes the derivatives incorrectly: the derivative of sin(4(2x‑1)) is 8 cos(8x‑4), not 4 cos(4(2x‑1)), and the derivative of 3(2x‑1) is 6, not 3. This makes the subsequent steps invalid.claude-sonnet-5-5: pass 2026-10-03 — The solution is correct. The factoring, the 0/0 L'Hôpital step, the derivative evaluation (8cos/6 reduced to 4cos/3), the substitution and the final simplification are all valid, and the rule labels fit.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic manipulation, L'Hopital's rule, and limit evaluation. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 applies L’Hôpital’s rule but incorrectly computes the derivatives: the derivative of sin(4(2x−1)) is 8 cos(8x−4), not 4 cos(4(2x−1)), and the derivative of 3(2x−1) is 6, not 3. This mis‑application leads to an incorrect intermediate expression.qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the indeterminate form, applies L'Hopital's rule, and simplifies the resulting expression to the correct limit. Each step applies a single rule and is labeled correctly.gpt-oss:20b: fail (error) 2026-09-29 — Step 3 applies L’Hôpital’s rule but the derivatives are incorrect: the derivative of sin(4(2x‑1)) is 8 cos(4(2x‑1)), not 4 cos(4(2x‑1)), and the derivative of 3(2x‑1) is 6, not 3. This makes the subsequent steps invalid.
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 2026-10-03 with SymPy 1.14.0.