Limit of \( \displaystyle \frac{2 x - \sin{\left(2 x - 1 \right)} - 1}{\left(2 x - 1\right)^{3}} \) as \( x \to \frac{1}{2} \)
Problem 1.718 · hard Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{2 x - \sin{\left(2 x - 1 \right)} - 1}{\left(2 x - 1\right)^{3}} \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{2 x - \sin{\left(2 x - 1 \right)} - 1}{\left(2 x - 1\right)^{3}}\right) \]limit algebraEvaluate the limit of the given function. Rearrange the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(2 x - \sin{\left(2 x - 1 \right)} - 1\right)}{\frac{d}{d x} \left(2 x - 1\right)^{3}}\right) \]lhopitalApply L'Hopital's Rule.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 - 2 \cos{\left(2 x - 1 \right)}}{6 \left(2 x - 1\right)^{2}}\right) \]simplify simplifyCompute the derivatives. Simplify the denominator.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{1 - \cos{\left(2 x - 1 \right)}}{3 \left(2 x - 1\right)^{2}}\right) \]algebraDivide numerator and denominator by 2.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(2 x - 1 \right)}\right)}{\frac{d}{d x} 3 \left(2 x - 1\right)^{2}}\right) \]lhopitalApply L'Hopital's Rule again.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{2 \sin{\left(2 x - 1 \right)}}{24 x - 12}\right) \]simplify simplifyCompute the derivatives. Simplify the denominator.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+}\left(\frac{\sin{\left(2 x - 1 \right)}}{12 x - 6}\right) \]algebraDivide numerator and denominator by 2.✓ Proved
- \[ = \frac{\lim_{x \to \frac{1}{2}^+}\left(\frac{\sin{\left(2 x - 1 \right)}}{2 x - 1}\right)}{6} \]limit-lawSplit the limit using the product rule.✓ Proved
- \[ = \frac{\lim_{x \to \frac{1}{2}^+} \cos{\left(2 x - 1 \right)}}{6} \]lhopital algebraApply L'Hopital's Rule to the remaining limit. Simplify the fraction.✓ Proved
- \[ = \frac{1}{6} \]substitution limitSubstitute x = 1/2. Evaluate the final value.✓ Proved
Answer \( \frac{1}{6} \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt. 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 2*x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where Derivative((2*x - 1)**3, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative((2*x - 1)**3, x) = 0 undefined where 2*x - 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 undefined where Derivative(3*(2*x - 1)**2, x) = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(3*(2*x - 1)**2, x) = 0 undefined where 24*x - 12 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 24*x - 12 = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 24*x - 12 = 0 undefined where 12*x - 6 = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 12*x - 6 = 0 undefined where 2*x - 1 = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 1 = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 15 | ✓ 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) — Steps 3, 7, and 12 are labeled 'lhopital' but perform differentiation, which is a separate operation from the rule application itself; the differentiation should be a distinct step labeled 'derivative' or 'simplify', and the application of L'Hôpital's rule should be a separate step. This violates the 'one rule per step' constraint.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-19 — Steps 3, 7, and 12 are labeled 'lhopital' but perform differentiation, which is a separate operation from the rule application itself; the differentiation should be a distinct step labeled 'derivative' or 'simplify', and the application of L'Hôpital's rule should be a separate step. This violates the 'one rule per step' constraint.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly applies L'Hopital's Rule and algebraic simplifications at each step. Each step changes only one aspect of the expression, and the labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-18 — Step 12 incorrectly applies L'Hopital's Rule to the expression sin(2*x - 1) / (6*(2*x - 1)). L'Hopital's Rule requires differentiating the numerator and denominator separately; the step instead differentiates the entire fraction (applying the quotient rule) and incorrectly drops the denominator's derivative terms, resulting in 2*cos(2*x - 1) / 2 rather than the correct derivative ratio 2*cos(2*x - 1) / 12. This is a fundamental misuse of the rule.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.