Limit of \( \displaystyle \frac{1 - \cos{\left(3 x + 1 \right)}}{3 x + 1} \) as \( x \to - \frac{1}{3} \)
Problem 1.810 · medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{1 - \cos{\left(3 x + 1 \right)}}{3 x + 1} \).
- \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{1 - \cos{\left(3 x + 1 \right)}}{3 x + 1}\right) \]limitStarting with the limit of the given function.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+} \text{NaN} \]simplifyEvaluate the numerator and denominator at the point to check for indeterminate form.Not checked
- \[ = \lim_{x \to - \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(1 - \cos{\left(3 x + 1 \right)}\right)}{\frac{d}{d x} \left(3 x + 1\right)}\right) \]lhopitalApply L'Hôpital's rule since we have a 0/0 indeterminate form.Not checked
- \[ = \lim_{x \to - \frac{1}{3}^+} \sin{\left(3 x + 1 \right)} \]simplify simplifyCompute the derivatives of the numerator and denominator. Simplify the fraction by dividing the coefficients.✓ Proved
- \[ = 0 \]substitution algebra limitSubstitute the value x = -1/3 into the expression. Simplify the argument of the sine function. Evaluate the final value.✓ Proved
Answer \( 0 \)
Lines: 7 proved, 2 not checked. 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 | Not checked | — | simplify left nan; no point in the sample was defined on both lines undefined where 3*x + 1 = 0 |
| 3 | Not checked | — | simplify left nan; no point in the sample was defined on both lines 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 |
| 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 applies the rule "limit" to a constant value; the correct label would be "algebra" or "evaluate".qwen3.6:27b-mlx: pass
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-04gpt-oss:20b: fail (style) 2026-11-04 — Step 8 applies the rule "limit" to a constant value; the correct label would be "algebra" or "evaluate".qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly identifies the indeterminate form, applies L'Hôpital's rule, computes the derivatives, and simplifies to the final answer. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-03qwen3.6:27b-mlx: pass 2026-11-03 — The solution correctly identifies the indeterminate form, applies L'Hôpital's rule, computes the derivatives, and simplifies the resulting expression to find the limit. All steps adhere to the one-rule-per-step constraint and use valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-03
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 2026-11-04 with SymPy 1.14.0.