Limit of \( \displaystyle \frac{\left(3 x - 1\right)^{2} - 9}{3 x + 2} \) as \( x \to - \frac{2}{3} \)
Problem 1.811 · medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{2}{3}} \frac{\left(3 x - 1\right)^{2} - 9}{3 x + 2} \).
- \[ \lim_{x \to - \frac{2}{3}^+}\left(\frac{\left(3 x - 1\right)^{2} - 9}{3 x + 2}\right) \]limit algebraSet up the limit with the specified direction. Ensure the numerator and denominator are clearly grouped.✓ Proved
- \[ = \lim_{x \to - \frac{2}{3}^+}\left(\frac{9 x^{2} - 6 x - 8}{3 x + 2}\right) \]simplify simplifyExpand the squared binomial. Combine constant terms in the numerator.✓ Proved
- \[ = \lim_{x \to - \frac{2}{3}^+}\left(3 x - 4\right) \]factor cancelFactor the quadratic expression in the numerator. Cancel the common factor (3*x + 2).✓ Proved
- \[ = -6 \]limit algebra simplifyEvaluate the limit by direct substitution. Multiply the constant. Final result.✓ Proved
Answer \( -6 \)
✓ 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 + 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 2 = 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 |
| 9 | ✓ 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 algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct substitution. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-11-06 — The solution correctly applies algebraic simplification, factoring, and cancellation to resolve the indeterminate form, followed by direct substitution. Each step adheres to the single-rule constraint and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-06qwen3.6:27b-mlx: pass 2026-11-05 — The solution correctly applies algebraic simplification, factoring, and cancellation to evaluate the limit. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: pass 2026-11-05qwen3.6:27b-mlx: pass 2026-11-05gpt-oss:20b: pass 2026-11-05
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 2026-11-06 with SymPy 1.14.0.