Limit of \( \displaystyle \frac{3 x - \sin{\left(3 x + 1 \right)} + 1}{\left(3 x + 1\right)^{3}} \) as \( x \to - \frac{1}{3} \)
Problem 1.773 · hard Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{3 x - \sin{\left(3 x + 1 \right)} + 1}{\left(3 x + 1\right)^{3}} \).
- \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{3 x - \sin{\left(3 x + 1 \right)} + 1}{\left(3 x + 1\right)^{3}}\right) \]limitIdentify the limit to evaluate.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(3 x - \sin{\left(3 x + 1 \right)} + 1\right)}{\frac{d}{d x} \left(3 x + 1\right)^{3}}\right) \]lhopitalThe limit is an indeterminate form 0/0, so apply L'Hopital's rule.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{3 - 3 \cos{\left(3 x + 1 \right)}}{9 \left(3 x + 1\right)^{2}}\right) \]simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the expression in the denominator.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{1 - \cos{\left(3 x + 1 \right)}}{3 \left(3 x + 1\right)^{2}}\right) \]algebraDivide the numerator and denominator by 3.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{2 \sin^{2}{\left(\frac{3 x}{2} + \frac{1}{2} \right)}}{3 \left(3 x + 1\right)^{2}}\right) \]rewrite substitutionUse the trigonometric identity 1 - cos(2theta) = 2*sin(theta)^2. Let u = 3*x + 1, so u approaches 0 as x approaches -1/3.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{2 \sin^{2}{\left(\frac{u}{2} \right)}}{3 u^{2}}\right) \]substitutionSubstitute u with 3*x + 1.✓ Proved
- \[ = \lim_{u \to 0^+} \frac{1}{6} \]trig-limit simplify simplifyUse the small angle approximation sin(theta) ~ theta for the limit. Expand the squared term. Cancel the u**2 terms and simplify the constant.✓ Proved
- \[ = \frac{1}{6} \]limitThe limit evaluates to the constant value.✓ Proved
Answer \( \frac{1}{6} \)
✓ 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. How we verify
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 undefined where Derivative((3*x + 1)**3, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative((3*x + 1)**3, x) = 0 undefined where 3*x + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 1 = 0 undefined where u = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 12 | ✓ 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 L'Hopital's rule, simplifies the resulting expression, and uses standard trigonometric limits to evaluate the final constant. All steps adhere to the one-rule-per-step constraint and use valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-29 — The solution correctly applies L'Hopital's rule, simplifies the resulting expression, and uses standard trigonometric limits to evaluate the final constant. All steps adhere to the one-rule-per-step constraint and use valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-29qwen3.6:27b-mlx: fail (style) 2026-10-28 — Step 3 applies two rules at once: it computes the derivatives (which should be labeled 'derivative') and simplifies the resulting expression (labeled 'simplify'). The contract requires each step to change only one thing.gpt-oss:20b: pass 2026-10-28qwen3.6:27b-mlx: pass 2026-10-28 — The solution correctly applies L'Hopital's rule, simplifies the resulting expression, and uses standard trigonometric identities and limit laws to arrive at the correct answer. Each step adheres to the single-rule constraint and uses valid labels from the provided vocabulary.gpt-oss:20b: pass 2026-10-28
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 2026-10-29 with SymPy 1.14.0.