Limit of \( \displaystyle \frac{3 \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 4} \) as \( x \to \infty \)
Problem 1.42 · hard Video
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 4} \).
- \[ \lim_{x \to \infty}\left(\frac{3 \left(2 x - 1\right)^{2} + 1}{\left(2 x - 1\right)^{2} - 4}\right) \]limitEvaluate the limit of the given function as x approaches infinity.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{12 x^{2} - 12 x + 4}{4 x^{2} - 4 x - 3}\right) \]algebra simplifyExpand the squared binomials. Distribute the 3 and combine constant terms.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{12 x^{2}}{4 x^{2} - 4 x - 3} - \frac{12 x}{4 x^{2} - 4 x - 3} + \frac{4}{4 x^{2} - 4 x - 3}\right) \]limit-lawSplit the fraction into three separate parts.✓ Proved
- \[ = - \lim_{x \to \infty}\left(\frac{12 x}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{12 x^{2}}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{4 x^{2} - 4 x - 3}\right) \]limit-lawApply the limit law for sums and differences.✓ Proved
- \[ = \lim_{x \to \infty} 3 - \lim_{x \to \infty}\left(\frac{12 x}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{4 x^{2} - 4 x - 3}\right) \]simplifyDivide the numerator and denominator of the first term by x**2.✓ Proved
- \[ = 3 - \lim_{x \to \infty}\left(\frac{12 x}{4 x^{2} - 4 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{4 x^{2} - 4 x - 3}\right) \]simplifySimplify the first limit term.✓ Proved
- \[ = 3 \]limit simplifyEvaluate the remaining limits as x approaches infinity. Final result.✓ Proved
Answer \( 3 \)
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. A reviewer's objection to the explanation stands; see the receipt. 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 (2*x - 1)**2 - 4 = 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 4*x**2 - 4*x - 3 = 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: fail (style) — Step 4 incorrectly labels an algebraic split of a fraction as a limit-law. The operation is purely algebraic and should be labeled "algebra" (or "simplify").qwen3.6:27b-mlx: fail (error) — Step 6 is mathematically incorrect. It claims to simplify `12*x**2 / (4*x**2 - 4*x - 3)` to `12/4` by dividing by `x**2`, but fails to divide the denominator terms by `x**2` (which would yield `4 - 4/x - 3/x**2`). The step effectively drops the lower-order terms without justification or correct algebraic transformation, resulting in an invalid equality between the expressions in steps 5 and 6.
Senior review claude-sonnet-5-5, 2026-10-03: fail — The final answer 3 is correct and every line is SymPy-equivalent. Step 4 is mislabeled (algebra, not limit-law). Step 6 combines division by x**2 with limit evaluation of the denominator and labels the result 'simplify', which misleads a student about how 12/4 arises.
gpt-oss:20b: uphold — Step 4 only splits one fraction into three over a common denominator, which is algebra/rewrite; no limit law is applied until step 5, so the 'limit-law' label is wrong.qwen3.6:27b-mlx: uphold — Dividing by x**2 would give 12/(4 - 4/x - 3/x**2), not 12/4; reaching 12/4 also needs a quotient limit law applied to the denominator terms, so the step does two things and 'simplify' misnames it.(a panel model has both passed and objected to this solution under the same prompt): dismiss — This is not a specific objection, so there is no defect to uphold.
Every verdict on record (7)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — Step 6 is mathematically incorrect. It claims to simplify `12*x**2 / (4*x**2 - 4*x - 3)` to `12/4` by dividing by `x**2`, but fails to divide the denominator terms by `x**2` (which would yield `4 - 4/x - 3/x**2`). The step effectively drops the lower-order terms without justification or correct algebraic transformation, resulting in an invalid equality between the expressions in steps 5 and 6.gpt-oss:20b: fail (style) 2026-10-03 — Step 4 incorrectly labels an algebraic split of a fraction as a limit-law. The operation is purely algebraic and should be labeled "algebra" (or "simplify").claude-sonnet-5-5: fail (misleading) 2026-10-03 — The final answer 3 is correct and every line is SymPy-equivalent. Step 4 is mislabeled (algebra, not limit-law). Step 6 combines division by x**2 with limit evaluation of the denominator and labels the result 'simplify', which misleads a student about how 12/4 arises.qwen3.6:27b-mlx: fail (error) 2026-09-29 — Step 6 claims to simplify the first term by dividing by x**2, but the result shown (12/4) is the limit of that term, not an algebraic simplification of the expression inside the Limit operator. The step conflates algebraic manipulation with limit evaluation, violating the 'one rule per step' constraint and mislabeling a limit evaluation as 'simplify'.gpt-oss:20b: fail (style) 2026-09-29 — Step 4 incorrectly labels an algebraic rewrite as a limit‑law application. Splitting the fraction into three terms is a single algebraic manipulation, not a limit law. The label should be "algebra" (or "rewrite"), not "limit-law".qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies limit laws and algebraic simplifications. Each step changes only one aspect of the expression or applies a single rule, and the labels are appropriate.gpt-oss:20b: fail (style) 2026-09-29 — Step 4 incorrectly labels the algebraic split of a fraction as a limit‑law. The operation is a single algebraic manipulation, not a limit law. This mislabeling could mislead a student about the appropriate rule to apply.
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.