Limit of \( \displaystyle \frac{3 \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 4} \) as \( x \to \infty \)
Problem 1.416 · hard
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{3 \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 4} \).
- \[ \lim_{x \to \infty}\left(\frac{3 \left(3 x - 1\right)^{2} + 1}{\left(3 x - 1\right)^{2} - 4}\right) \]limitStart with the original limit.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{27 x^{2} - 18 x + 4}{9 x^{2} - 6 x - 3}\right) \]algebra simplifyExpand the squared terms. Simplify the numerator and denominator.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{27 x^{2}}{9 x^{2} - 6 x - 3} - \frac{18 x}{9 x^{2} - 6 x - 3} + \frac{4}{9 x^{2} - 6 x - 3}\right) \]limit-lawSplit the fraction into three parts.✓ Proved
- \[ = \lim_{x \to \infty}\left(- \frac{18 x}{9 x^{2} - 6 x - 3}\right) + \lim_{x \to \infty}\left(\frac{27 x^{2}}{9 x^{2} - 6 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{9 x^{2} - 6 x - 3}\right) \]limit-lawApply the sum rule for limits.✓ Proved
- \[ = \lim_{x \to \infty} 3 + \lim_{x \to \infty}\left(- \frac{18 x}{9 x^{2} - 6 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{9 x^{2} - 6 x - 3}\right) \]simplifyDivide the leading terms of the first fraction.✓ Proved
- \[ = 3 + \lim_{x \to \infty}\left(- \frac{18 x}{9 x^{2} - 6 x - 3}\right) + \lim_{x \to \infty}\left(\frac{4}{9 x^{2} - 6 x - 3}\right) \]simplifySimplify the first term.✓ Proved
- \[ = 3 + \lim_{x \to \infty}\left(\frac{4}{9 x^{2} - 6 x - 3}\right) \]limit-lawEvaluate the second limit.✓ Proved
- \[ = 3 \]limit-law simplifyEvaluate the third limit. 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.
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 - 1)**2 - 4 = 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 - 6*x - 3 = 0 |
| 10 | ✓ 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 4 applies a limit-law to split a single fraction into three separate fractions, which is not a valid application of the limit law. The correct approach is to perform the algebraic split first, then apply limit laws to each term. This step combines two distinct operations (algebraic manipulation and limit law) in one line, violating the contract.qwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-08: fail — The answer 3 is correct. Step 6 changes the function inside the limit to a non-equal expression under the label 'simplify', and the proper route is to divide numerator and denominator by x^2 and then apply limit laws. Step 4 is also mislabeled as 'limit-law' when it is an algebraic split.
gpt-oss:20b: uphold — Step 4 only splits one fraction into three, which is algebra, so the label 'limit-law' names a rule the step did not apply. The claim about step 6 is inaccurate, since step 6 is labeled 'simplify' and not 'limit-law', but the step 4 mislabel is a real defect.qwen3.6:27b-mlx: uphold — Step 6 replaces 27x^2/(9x^2-6x-3) with the constant 27/9 inside the limit. These are not equal functions and 'simplify' cannot turn one into the other. Only the limit values agree, so the step is a leading-term shortcut mislabeled as simplification and would teach a false rule. The final value is still correct, so this is misleading rather than an outright math error.
Every verdict on record (5)
gpt-oss:20b: fail (error) 2026-10-08 — Step 4 applies a limit-law to split a single fraction into three separate fractions, which is not a valid application of the limit law. The correct approach is to perform the algebraic split first, then apply limit laws to each term. This step combines two distinct operations (algebraic manipulation and limit law) in one line, violating the contract.qwen3.6:27b-mlx: pass 2026-10-08claude-sonnet-5-5: fail (misleading) 2026-10-08 — The answer 3 is correct. Step 6 changes the function inside the limit to a non-equal expression under the label 'simplify', and the proper route is to divide numerator and denominator by x^2 and then apply limit laws. Step 4 is also mislabeled as 'limit-law' when it is an algebraic split.qwen3.6:27b-mlx: fail (error) 2026-10-08 — Step 6 is mathematically incorrect. It claims that Limit(27*x**2 / (9*x**2 - 6*x - 3), x, oo) equals Limit(27/9, x, oo). The limit of the ratio of leading terms is 3, but the limit of the constant 27/9 is also 3. However, the step implies that the expression inside the limit *becomes* 27/9, which is false (the expression is a function of x, not a constant). More critically, Step 6 applies 'simplify' to change the integrand of a limit, but it does so by dropping lower order terms without justification in that single step, or it incorrectly asserts equality between the function and its leading coefficient ratio. Actually, looking closer: Step 6 changes `27*x**2 / (9*x**2 - 6*x - 3)` to `27/9`. This is not an algebraic equality. The expressions are not equal. The limit of the first is 3, the limit of the second is 3. But the step says `Limit(A, ...) = Limit(B, ...)` where A != B. This is a defect because the step changes the expression inside the limit to a non-equal expression. The rule 'simplify' usually implies algebraic simplification (equality). Replacing a rational function with its leading coefficient ratio is not an algebraic simplification of the expression itself, but an evaluation of the limit. If it were an evaluation, the Limit wrapper should disappear or the result should be the number. Keeping the Limit wrapper around a changed, non-equal expression is an error.gpt-oss:20b: fail (style) 2026-10-08 — Step 4 incorrectly labels an algebraic split of a single fraction as a limit-law. Step 6 also mislabels the simplification of the leading terms as a limit-law. These labeling errors violate the contract that each step must apply exactly one rule from the permitted vocabulary.
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 2026-10-08 with SymPy 1.14.0.