Limit of \( \displaystyle \frac{\cos{\left(2 x - 1 \right)}}{2 x} \) as \( x \to \frac{1}{2} \)
Problem 1.82 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{2}} \frac{\cos{\left(2 x - 1 \right)}}{2 x} \).
- \[ \lim_{x \to \frac{1}{2}^+}\left(\frac{\cos{\left(2 x - 1 \right)}}{2 x}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to \frac{1}{2}^+} \cos{\left(2 x - 1 \right)} \left(\lim_{x \to \frac{1}{2}^+}\left(2 x\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = 1 \]limit algebra simplify simplifyEvaluate the limits of the numerator and denominator separately. Simplify the expression inside the numerator and denominator. Evaluate the cosine of zero. Final result.✓ Proved
Answer \( 1 \)
✓ 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 x = 0 undefined where Limit(2*x, x, 1/2, dir='+') = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(2*x, x, 1/2, dir='+') = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the quotient law for limits and evaluates the resulting expressions step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the quotient law for limits and evaluates the resulting expressions step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies the quotient law for limits and evaluates the resulting expressions step-by-step. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-30
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.