Limit of \( \displaystyle \frac{1 - \cos{\left(3 x - 1 \right)}}{3 x - 1} \) as \( x \to \frac{1}{3} \)
Problem 1.502 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{1 - \cos{\left(3 x - 1 \right)}}{3 x - 1} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{1 - \cos{\left(3 x - 1 \right)}}{3 x - 1}\right) \]limitStart with the limit of the given function.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(3 x - 1 \right)}\right)}{\frac{d}{d x} \left(3 x - 1\right)}\right) \]lhopitalApply L'Hopital's rule because it is an indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+} \sin{\left(3 x - 1 \right)} \]simplify simplifyDifferentiate the numerator and denominator. Simplify the fraction.✓ Proved
- \[ = 0 \]substitution algebra simplify limitSubstitute the value x = 1/3 into the expression. Simplify the expression inside the sine function. Evaluate the expression inside the sine. The final value of the limit is 0.✓ Proved
Answer \( 0 \)
✓ 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 3*x - 1 = 0 undefined where Derivative(3*x - 1, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(3*x - 1, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 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: fail (style) — Step 8 incorrectly labels the final evaluation as a "limit" step; after substitution the expression is a constant, so the correct label would be "simplify" or "evaluate".qwen3.6:27b-mlx: pass — The solution correctly applies L'Hopital's rule and simplifies the resulting expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: fail (style) 2026-10-10 — Step 8 incorrectly labels the final evaluation as a "limit" step; after substitution the expression is a constant, so the correct label would be "simplify" or "evaluate".qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies L'Hopital's rule and simplifies the resulting expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.qwen3.6:27b-mlx: pass 2026-10-10 — The solution correctly applies L'Hopital's rule and simplifies the resulting expression. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-10
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 2026-10-10 with SymPy 1.14.0.