Limit of \( \displaystyle \frac{- 10 x + \left(2 x + 1\right)^{2} + 1}{2 x - 2} \) as \( x \to 1 \)
Problem 1.91 · medium
Evaluate \( \displaystyle \lim_{x \to 1} \frac{- 10 x + \left(2 x + 1\right)^{2} + 1}{2 x - 2} \).
- \[ \lim_{x \to 1^+}\left(\frac{- 10 x + \left(2 x + 1\right)^{2} + 1}{2 x - 2}\right) \]limitStart with the limit of the given function.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{4 x^{2} - 6 x + 2}{2 x - 2}\right) \]algebra simplify factorExpand the squared term. Combine like terms in the numerator. Factor out a 2 from the numerator.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{\left(x - 1\right) \left(4 x - 2\right)}{2 x - 2}\right) \]factorFactor the quadratic expression in the numerator.✓ Proved
- \[ = \lim_{x \to 1^+}\left(\frac{2 x - 1}{x - 1}\right) \]cancelCancel the common factor (x - 1).Not checked
- \[ = \lim_{x \to 1^+} 1 \]limitEvaluate the remaining expression by substituting x = 1.Not checked
- \[ = 1 \]limitThe limit is 1.✓ Proved
Answer \( 1 \)
Lines: 7 proved, 2 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
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 2*x - 2 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*x - 2 = 0 |
| 6 | Not checked | — | simplify left -oo; no point in the sample was defined on both lines undefined where 2*x - 2 = 0 undefined where x - 1 = 0 |
| 7 | Not checked | — | simplify left oo; no point in the sample was defined on both lines undefined where x - 1 = 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 (error) — Step 6 cancels only the factor (x-1) but leaves a remaining denominator of (x-1); the factor 2 in the numerator and denominator also cancel, so the correct simplification is (2*x-1), not (2*x-1)/(x-1).qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 6 incorrectly cancels the factor of 2 along with (x-1). The expression in Step 5 is 2*(2*x - 1)*(x - 1) / (2*(x - 1)), which simplifies to (2*x
Every verdict on record (4)
qwen3.6:27b-mlx: inconclusive 2026-10-03 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 6 incorrectly cancels the factor of 2 along with (x-1). The expression in Step 5 is 2*(2*x - 1)*(x - 1) / (2*(x - 1)), which simplifies to (2*xgpt-oss:20b: fail (error) 2026-10-03 — Step 6 cancels only the factor (x-1) but leaves a remaining denominator of (x-1); the factor 2 in the numerator and denominator also cancel, so the correct simplification is (2*x-1), not (2*x-1)/(x-1).qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 6 incorrectly cancels the factor of 2 from the numerator and denominator. The expression in Step 5 is 2*(2*x - 1)*(x - 1) / (2*(x - 1)). Canceling (x-1) leaves 2*(2*x - 1) / 2, which simplifies to (2*x - 1). The solver dropped the factor of 2 from the numerator, resulting in (2*x - 1)/(x - 1) instead of the correct (2*x - 1). This leads to the wrong final answer of 1 instead of 1.gpt-oss:20b: fail (error) 2026-09-30 — Step 6 incorrectly cancels the common factor (x-1); the expression should reduce to 2*x-1, not (2*x-1)/(x-1). Consequently, step 7 is also incorrect.
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.