Limit of \( \displaystyle \frac{\sin{\left(4 x \right)}}{3 x} \) as \( x \to 0 \)
Problem 1.152 · easy
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sin{\left(4 x \right)}}{3 x} \).
- \[ \lim_{x \to 0^+}\left(\frac{\sin{\left(4 x \right)}}{3 x}\right) \]limit algebra algebraStart with the original limit. Factor out the constant 1/3. Multiply and divide by 4 to use the fundamental trig limit.✓ Proved
- \[ = \frac{4 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{3} \]substitutionLet u = 4x, so u approaches 0 as x approaches 0.✓ Proved
- \[ = \frac{4}{3} \]trig-limit simplifyApply the standard limit sin(u)/u -> 1. Final result.✓ Proved
Answer \( \frac{4}{3} \)
✓ 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 |
| 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 undefined where u = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 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 algebraic manipulation, substitution, and the standard trigonometric limit. 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 2026-10-04 — The solution correctly applies algebraic manipulation, substitution, and the standard trigonometric limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies algebraic manipulation to isolate the standard trigonometric limit form, uses substitution appropriately, and evaluates the limit accurately. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-04
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.