Limit of \( \displaystyle \frac{9 x^{2} - 15 x + 6}{3 x - 3} \) as \( x \to 1 \)
Problem 1.146 · medium
Evaluate \( \displaystyle \lim_{x \to 1} \frac{9 x^{2} - 15 x + 6}{3 x - 3} \).
- \[ \lim_{x \to 1^+}\left(\frac{9 x^{2} - 15 x + 6}{3 x - 3}\right) \]limit factorState the initial limit problem. Factor out 3 from the numerator.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(9 x - 6\right)}{3 x - 3}\right) \]factor factorFactor the quadratic expression in the numerator. Factor out 3 from the denominator.✓ Proved
- \[ = \lim_{x \to 1^+}\left(3 x - 2\right) \]algebra cancelCancel the common factor of 3. Cancel the common factor (x - 1).✓ Proved
- \[ = 1 \]limitEvaluate the limit by substituting x = 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 - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 3 = 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 |
| 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 5 applies a cancellation of the factor (x-1) but labels it "algebra"; it should be labeled "cancel". Step 6 then incorrectly labels the substitution of the limit as "cancel"; it should be "limit". These mislabelings violate the rule that each step must name the exact rule applied.qwen3.6:27b-mlx: pass — The solution correctly factors the numerator and denominator, cancels common terms, and evaluates the limit. Each step applies a single rule from the allowed vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly factors the numerator and denominator, cancels common terms, and evaluates the limit. Each step applies a single rule from the allowed vocabulary.gpt-oss:20b: fail (style) 2026-10-04 — Step 5 applies a cancellation of the factor (x-1) but labels it "algebra"; it should be labeled "cancel". Step 6 then incorrectly labels the substitution of the limit as "cancel"; it should be "limit". These mislabelings violate the rule that each step must name the exact rule applied.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly factors the numerator and denominator, cancels common factors, and evaluates the limit. Each step applies a single rule from the allowed vocabulary.gpt-oss:20b: fail (style) 2026-10-04 — Step 5’s note says "Cancel the common factor of 3" but the cancellation performed is of the factor (x‑1). The note should reflect the actual operation.
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 2026-10-04 with SymPy 1.14.0.