Limit of \( \displaystyle \frac{\sin{\left(9 x + 3 \right)}}{3 x + 1} \) as \( x \to - \frac{1}{3} \)
Problem 1.425 · medium
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\sin{\left(9 x + 3 \right)}}{3 x + 1} \).
- \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sin{\left(9 x + 3 \right)}}{3 x + 1}\right) \]limit factorEvaluate the limit of the function as x approaches -1/3. Factor the argument of the sine function.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \sin{\left(9 x + 3 \right)}}{\frac{d}{d x} \left(3 x + 1\right)}\right) \]lhopitalApply L'Hopital's Rule because the limit is an indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(3 \cos{\left(9 x + 3 \right)}\right) \]simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the expression by canceling the common factor of 3.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+} 3 \]substitutionSubstitute x = -1/3 into the expression.✓ Proved
- \[ = 3 \]limit simplify simplifyEvaluate the limit. Evaluate the cosine of 0. Final result.✓ 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. 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 3*x + 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where Derivative(3*x + 1, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(3*x + 1, 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 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ 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 4 is mathematically incorrect. The derivative of the numerator sin(3*(3x+1)) with respect to x is 9*cos(3*(3x+1)), but the step writes (3*cos(3*(3x+1))*3), which implies a chain rule application of 3*3, but the structure suggests a confusion in applying the chain rule or simplifying prematurely. More critically, Step 4 claims to 'Compute the derivatives' but the expression (3*cos(3*(3*x + 1))*3)/3 is not the direct result of differentiating sin(3*(3x+1)) and 3x+1; the derivative of sin(3*(3x+1)) is 9cos(3*(3x+1)). While 9/3 simplifies to 3, the step label 'simplify' is used for a step that performs differentiation (which should be labeled 'derivative' or part of 'lhopital' if combined, but here lhopital was step 3). Actually, looking closely: Step 3 applies L'Hopital's rule, resulting in a limit of a ratio of derivatives. Step 4 computes those derivatives. The derivative of sin(3*(3x+1)) is cos(3*(3x+1)) * 3 * 3 = 9cos(...). The step writes (3*cos(...)*3)/3. This is algebraically equal to 3cos(...). However, the label is 'simplify'. Computing derivatives is not 'simplify'. It is 'derivative'. Furthermore, Step 3 to Step 4 changes the expression from a symbolic limit of derivatives to the computed derivatives. This is a 'derivative' step, not 'simplify'. Using 'simplify' to label the computation of derivatives is a defect in labeling.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 4 is mathematically incorrect. The derivative of the numerator sin(3*(3x+1)) with respect to x is 9*cos(3*(3x+1)), but the step writes (3*cos(3*(3x+1))*3), which implies a chain rule application of 3*3, but the structure suggests a confusion in applying the chain rule or simplifying prematurely. More critically, Step 4 claims to 'Compute the derivatives' but the expression (3*cos(3*(3*x + 1))*3)/3 is not the direct result of differentiating sin(3*(3x+1)) and 3x+1; the derivative of sin(3*(3x+1)) is 9cos(3*(3x+1)). While 9/3 simplifies to 3, the step label 'simplify' is used for a step that performs differentiation (which should be labeled 'derivative' or part of 'lhopital' if combined, but here lhopital was step 3). Actually, looking closely: Step 3 applies L'Hopital's rule, resulting in a limit of a ratio of derivatives. Step 4 computes those derivatives. The derivative of sin(3*(3x+1)) is cos(3*(3x+1)) * 3 * 3 = 9cos(...). The step writes (3*cos(...)*3)/3. This is algebraically equal to 3cos(...). However, the label is 'simplify'. Computing derivatives is not 'simplify'. It is 'derivative'. Furthermore, Step 3 to Step 4 changes the expression from a symbolic limit of derivatives to the computed derivatives. This is a 'derivative' step, not 'simplify'. Using 'simplify' to label the computation of derivatives is a defect in labeling.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly applies L'Hopital's Rule after factoring the argument, and all subsequent algebraic simplifications and evaluations are correct.gpt-oss:20b: pass 2026-10-08
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 2026-10-08 with SymPy 1.14.0.