Limit of \( \displaystyle \frac{\cos{\left(3 x + 1 \right)}}{3 x + 2} \) as \( x \to - \frac{1}{3} \)
Problem 1.73 · medium
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\cos{\left(3 x + 1 \right)}}{3 x + 2} \).
- \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\cos{\left(3 x + 1 \right)}}{3 x + 2}\right) \]limitStart with the original limit expression.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+} 1 \]substitution algebra simplifySubstitute the value x = -1/3 into the expression to check for indeterminate forms. Simplify the terms inside the cosine and the denominator. Evaluate the constants.✓ Proved
- \[ = 1 \]limitThe limit evaluates to 1.✓ Proved
Answer \( 1 \)
✓ 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 + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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 |
| 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 direct substitution to evaluate the limit, as the function is continuous at the point of interest. Each step changes only one aspect of the expression and uses valid labels.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies direct substitution to evaluate the limit, as the function is continuous at the point of interest. Each step changes only one aspect of the expression and uses valid labels.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-30 — The solution correctly identifies that the limit can be evaluated by direct substitution since the function is continuous at x = -1/3. Each step applies a single rule and uses labels from the allowed vocabulary.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.