Limit of \( \displaystyle \frac{\cos{\left(2 x \right)}}{2 x + 1} \) as \( x \to 0 \)
Problem 1.112 · medium
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\cos{\left(2 x \right)}}{2 x + 1} \).
- \[ \lim_{x \to 0^+}\left(\frac{\cos{\left(2 x \right)}}{2 x + 1}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = 1 \]limit simplify simplifySubstitute the value x = 0 into the expression. Simplify the numerator and denominator. Final evaluation.✓ 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 + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 4 | ✓ 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 2 applies a substitution of x=0 into the expression, but it is labeled as "limit". The correct rule name from the allowed vocabulary is "substitution". This mislabeling is a stylistic defect.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: fail (style) 2026-10-03 — Step 2 applies a substitution of x=0 into the expression, but it is labeled as "limit". The correct rule name from the allowed vocabulary is "substitution". This mislabeling is a stylistic defect.qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03
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.