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 1.797 · hard Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{3 x - \sin{\left(3 x - 1 \right)} - 1}{\left(3 x - 1\right)^{3}} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{3 x - \sin{\left(3 x - 1 \right)} - 1}{\left(3 x - 1\right)^{3}}\right) \]limit algebraEvaluate the limit of the given function at x = 1/3. Group the terms in the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(3 x - \sin{\left(3 x - 1 \right)} - 1\right)}{\frac{d}{d x} \left(3 x - 1\right)^{3}}\right) \]lhopitalApply L'Hôpital's rule since the limit is an indeterminate form 0/0.✓ 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 simplify algebraCompute the derivatives in the numerator and denominator. Simplify the denominator. Factor out 3 from the numerator.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{1 - \cos{\left(3 x - 1 \right)}}{3 \left(3 x - 1\right)^{2}}\right) \]simplify rewriteSimplify the fraction. Rewrite the expression to isolate the term (x - 1/3).✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{1 - \cos{\left(3 u \right)}}{27 u^{2}}\right) \]substitution simplifySubstitute u = x - 1/3, which implies u approaches 0 as x approaches 1/3. Simplify the denominator (3 * (3*u)**2 = 27*u**2).✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{\frac{d}{d u} \left(1 - \cos{\left(3 u \right)}\right)}{\frac{d}{d u} 27 u^{2}}\right) \]lhopitalApply L'Hôpital's rule again.✓ Proved
- \[ = \lim_{u \to 0^+}\left(\frac{\sin{\left(3 u \right)}}{18 u}\right) \]simplify algebraCompute the derivatives. Simplify the coefficients.✓ Proved
- \[ = \lim_{u \to 0^+} \frac{1}{6} \]trig-limitUse the standard limit sin(kz)/z = k as z approaches 0.✓ Proved
- \[ = \frac{1}{6} \]limitFinal evaluation of the limit.✓ 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 |
| 3 | ✓ 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 |
| 4 | ✓ 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 |
| 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 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 undefined where u = 0 |
| 10 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 11 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 undefined where Derivative(27*u**2, u) = 0 |
| 12 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(27*u**2, u) = 0 undefined where u = 0 |
| 13 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 14 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where u = 0 |
| 15 | ✓ 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'Hôpital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-02 — The solution correctly applies L'Hôpital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-02qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies L'Hôpital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-01qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly applies L'Hôpital's rule and algebraic simplifications. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-01
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 2026-11-02 with SymPy 1.14.0.