Limit of \( \displaystyle \frac{\sin{\left(15 x \right)}}{\sin{\left(6 x \right)}} \) as \( x \to 0 \)
Problem 1.80 · medium
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\sin{\left(15 x \right)}}{\sin{\left(6 x \right)}} \).
- \[ \lim_{x \to 0^+}\left(\frac{\sin{\left(15 x \right)}}{\sin{\left(6 x \right)}}\right) \]limitIdentify the limit to be evaluated.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(15 x \right)}}{\frac{d}{d x} \sin{\left(6 x \right)}}\right) \]lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's Rule.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{5 \cos{\left(15 x \right)}}{2 \cos{\left(6 x \right)}}\right) \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = \frac{5}{2} \]limit simplify simplifyEvaluate the limit by substituting x = 0. Simplify the resulting fraction. Reduce the fraction to its simplest form.✓ Proved
Answer \( \frac{5}{2} \)
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. The reviewers disagree about how one step is explained; every verdict is in 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 sin(6*x) = 0 undefined where Derivative(sin(6*x), x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(sin(6*x), x) = 0 undefined where cos(6*x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where cos(6*x) = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: fail (error) — Step 3 applies the differentiation rule (derivative) to compute the derivatives, but is labeled 'simplify'. The label 'simplify' is reserved for presentational algebraic steps, not for performing differentiation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 3 applies the differentiation rule (derivative) to compute the derivatives, but is labeled 'simplify'. The label 'simplify' is reserved for presentational algebraic steps, not for performing differentiation.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly applies L'Hopital's Rule and simplifies the resulting expression. Each step changes only one aspect of the expression and uses an acceptable label from the fixed vocabulary.gpt-oss:20b: pass 2026-09-30
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.