Limit of \( \displaystyle \frac{\sin{\left(3 x \right)}}{x} \) as \( x \to 0 \)
Problem 1.15 · easy
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sin{\left(3 x \right)}}{x} \).
- \[ \lim_{x \to 0^+}\left(\frac{\sin{\left(3 x \right)}}{x}\right) \]limitIdentify the limit to evaluate.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{3 \sin{\left(x \right)}}{x}\right) \]rewrite algebraUse the substitution u = 3x, or rewrite the argument. Factor out the constant 3.✓ Proved
- \[ = \lim_{x \to 0^+} 3 \lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{x}\right) \]limit-lawApply the product rule for limits.✓ Proved
- \[ = 3 \]trig-limit simplifyApply the standard trigonometric limit sin(x)/x -> 1. Final evaluation.✓ Proved
Answer \( 3 \)
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 |
| 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(3*x)/x as 3*sin(x)/x. The identity sin(3x)=3sin(x) is false; the correct transformation would involve using the limit sin(3x)/(3x)→1 and then multiplying by 3, not replacing sin(3x) with 3sin(x).qwen3.6:27b-mlx: fail (error) — Step 2 is mathematically incorrect; sin(3x) is not equal to 3sin(x). The step attempts a substitution but fails to adjust the denominator, resulting in an invalid equality.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 2 is mathematically incorrect; sin(3x) is not equal to 3sin(x). The step attempts a substitution but fails to adjust the denominator, resulting in an invalid equality.gpt-oss:20b: fail (error) 2026-09-29 — Step 2 incorrectly rewrites sin(3*x)/x as 3*sin(x)/x. The identity sin(3x)=3sin(x) is false; the correct transformation would involve using the limit sin(3x)/(3x)→1 and then multiplying by 3, not replacing sin(3x) with 3sin(x).qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 2 incorrectly transforms sin(3x) into 3sin(x), which is mathematically false. The solver likely confused the substitution method (which would change the variable of integration/limit) with an invalid algebraic manipulation.gpt-oss:20b: fail (error) 2026-09-29 — Step 2 incorrectly rewrites sin(3*x)/x as 3*sin(x)/x, which is not an algebraic identity. The correct approach is to use the standard limit sin(3x)/(3x) → 1 and then multiply by 3, or rewrite as 3*(sin(3x)/(3x)).
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 2026-09-29 with SymPy 1.14.0.