Limit of \( \displaystyle \frac{\sin^{2}{\left(2 x \right)}}{4 x^{2}} \) as \( x \to 0 \)
Problem 1.117 · medium
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sin^{2}{\left(2 x \right)}}{4 x^{2}} \).
- \[ \lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(2 x \right)}}{4 x^{2}}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\sin^{2}{\left(x \right)}}{x^{2}}\right) \]algebra simplify algebraUse the double angle identity for sine, or simply expand the square. Simplify the constant coefficients. Rewrite the fraction as a square of a fraction.✓ Proved
- \[ = \left(\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right)\right)^{2} \]limit-lawApply the power rule for limits.✓ Proved
- \[ = 1 \]limit simplifyEvaluate the fundamental trigonometric limit. Final result.✓ Proved
Answer \( 1 \)
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. A reviewer's objection to the explanation stands; see the receipt.
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 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ 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 2 incorrectly rewrites sin(2*x)**2 as (sin(x)**2)*(2**2). The correct expansion is sin(2x)^2 = (2 sin x cos x)^2 = 4 sin^2 x cos^2 x, so the cos^2 term is missing. This violates the rule that each step must apply only one legitimate transformation.qwen3.6:27b-mlx: fail (error) — Step 2 incorrectly claims to use a double angle identity to transform sin(2x)^2 into 4sin(x)^2. The double angle identity is sin(2x) = 2sin(x)cos(x), which would yield 4sin^2(x)cos^2(x). The step actually applies the property sin(2x) = 2sin(x), which is false. The correct algebraic manipulation is sin(2x)^2 = (2sin(x)cos(x))^2 or simply recognizing the argument scaling, but the note explicitly cites the wrong identity.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 incorrectly claims to use a double angle identity to transform sin(2x)^2 into 4sin(x)^2. The double angle identity is sin(2x) = 2sin(x)cos(x), which would yield 4sin^2(x)cos^2(x). The step actually applies the property sin(2x) = 2sin(x), which is false. The correct algebraic manipulation is sin(2x)^2 = (2sin(x)cos(x))^2 or simply recognizing the argument scaling, but the note explicitly cites the wrong identity.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 incorrectly rewrites sin(2*x)**2 as (sin(x)**2)*(2**2). The correct expansion is sin(2x)^2 = (2 sin x cos x)^2 = 4 sin^2 x cos^2 x, so the cos^2 term is missing. This violates the rule that each step must apply only one legitimate transformation.qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 2 claims to use the double angle identity for sine, but the double angle identity is sin(2x) = 2sin(x)cos(x), not sin(2x) = 2sin(x). The step incorrectly replaces sin(2x) with 2sin(x), which is mathematically false.gpt-oss:20b: fail (error) 2026-10-03 — Step 2 incorrectly rewrites sin(2x)^2 as (sin x)^2·4, dropping the necessary cos^2 x factor. The algebraic transformation is invalid, so the subsequent steps are based on a false expression.
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.