Limit of \( \displaystyle \frac{\cos{\left(2 x + 1 \right)}}{2 x + 2} \) as \( x \to - \frac{1}{2} \)
Problem 1.172 · medium
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{2}} \frac{\cos{\left(2 x + 1 \right)}}{2 x + 2} \).
- \[ \lim_{x \to - \frac{1}{2}^+}\left(\frac{\cos{\left(2 x + 1 \right)}}{2 x + 2}\right) \]limit algebra algebra limitEvaluate the limit of the function as x approaches -1/2. Factor out the 2 in the denominator. Rewrite 2*x + 1 as 2*(x + 1/2). Check if the denominator is zero. Since 2*(-1/2) + 2 = 1, the limit can be evaluated by direct substitution.✓ Proved
- \[ = 1 \]limit simplify simplify simplifySubstitute x = -1/2 into the expression. Simplify the numerator and denominator. Evaluate cos(0). 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 2*x + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x + 2 = 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 |
| 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 (style) — Step 4 repeats the original expression without changing it and is labeled as a limit step. It neither applies a new rule nor alters the expression, violating the requirement that each step must change only one thing. This unnecessary duplication should be removed.qwen3.6:27b-mlx: pass — The solution correctly identifies that the limit can be evaluated by direct substitution since the denominator is non-zero at the limit point. The intermediate algebraic manipulations in steps 2 and 3 are unnecessary but harmless, and the final substitution is correct.
Senior review claude-sonnet-5-5, 2026-10-04: pass — The limit is correct: the function is continuous at x=-1/2, the denominator equals 1, and cos(0)=1. Lines 2-3 are redundant detours and line 4 reverts to the original expression, which is untidy but not mathematically wrong. SymPy proved every line.
gpt-oss:20b: dismiss — Step 4 is a legitimate use of the 'limit' rule: it restores the original form and notes the denominator is nonzero at x=-1/2 (2*(-1/2)+2=1), so direct substitution applies. Its redundancy is at most a style issue, not a false statement or a wrong rule name.
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies that the limit can be evaluated by direct substitution since the denominator is non-zero at the limit point. The intermediate algebraic manipulations in steps 2 and 3 are unnecessary but harmless, and the final substitution is correct.gpt-oss:20b: fail (style) 2026-10-04 — Step 4 repeats the original expression without changing it and is labeled as a limit step. It neither applies a new rule nor alters the expression, violating the requirement that each step must change only one thing. This unnecessary duplication should be removed.claude-sonnet-5-5: pass 2026-10-04 — The limit is correct: the function is continuous at x=-1/2, the denominator equals 1, and cos(0)=1. Lines 2-3 are redundant detours and line 4 reverts to the original expression, which is untidy but not mathematically wrong. SymPy proved every line.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies that the limit can be evaluated by direct substitution since the denominator is non-zero at x = -1/2. The intermediate algebraic manipulations are valid, and the steps adhere to the one-change-per-step rule.gpt-oss:20b: fail (error) 2026-10-04 — Step 4 reverts to the original expression and applies the "limit" rule incorrectly; it neither simplifies nor changes the expression and introduces a redundant, incorrect step. The solution should directly substitute after confirming the denominator is non‑zero, without this extra limit step.
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.