Limit of \( \displaystyle \frac{- 2 x + \left(2 x + 1\right)^{3} - 1}{2 \left(2 x + 1\right)^{3} + 5} \) as \( x \to \infty \)
Problem 1.302 · hard
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- 2 x + \left(2 x + 1\right)^{3} - 1}{2 \left(2 x + 1\right)^{3} + 5} \).
- \[ \lim_{x \to \infty}\left(\frac{- 2 x + \left(2 x + 1\right)^{3} - 1}{2 \left(2 x + 1\right)^{3} + 5}\right) \]limit simplifyStart with the original limit expression. Expand the cubic terms to see the polynomial structure.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{8 x^{3} + 12 x^{2} - 6 x}{16 x^{3} + 24 x^{2} + 16 x + 7}\right) \]simplifyThe expanded form shows the highest power of x.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{1 + \frac{3}{2 x} - \frac{3}{4 x^{2}}}{2 \left(1 + \frac{3}{2 x} + \frac{1}{x^{2}} + \frac{7}{16 x^{3}}\right)}\right) \]algebra algebraDivide the numerator and denominator by the highest power of x, which is x**3. Simplify the leading coefficient.✓ Proved
- \[ = \lim_{x \to \infty} \frac{1}{2} \]limitAs x approaches infinity, all terms with x in the denominator approach zero.✓ Proved
- \[ = \frac{1}{2} \]limitThe limit evaluates to the ratio of the leading coefficients.✓ Proved
Answer \( \frac{1}{2} \)
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 2*(2*x + 1)**3 + 5 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 2*(2*x + 1)**3 + 5 = 0 undefined where 16*x**3 + 24*x**2 + 16*x + 7 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**3 + 24*x**2 + 16*x + 7 = 0 undefined where 1 + 3/(2*x) + x**(-2) + 7/(16*x**3) = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 + 3/(2*x) + x**(-2) + 7/(16*x**3) = 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 1 + 3/(2*x) + x**(-2) + 7/(16*x**3) = 0 undefined where x = 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 (error) — Step 4 incorrectly rewrites the fraction by factoring out 8*x**3/16*x**3 and then mis‑expressing the remaining terms. The correct division by x**3 should yield (8+12/x-6/x**2)/(16+24/x+16/x**2+7/x**3).qwen3.6:27b-mlx: fail (error) — Step 2 is labeled 'simplify' but the note claims to expand terms, while the expression in Step 2 is identical to Step 1 (no expansion occurred). Step 3 performs the expansion, but the label 'simplify' is vague for an expansion step, and the transition from Step 2 to Step 3 applies the expansion rule without a proper label like 'algebra' or 'expand' (though 'simplify' is in the vocab, the previous step was a no-op). More critically, Step 4 applies 'algebra' to divide by x^3, but the expression shown is already divided and factored, implying the division happened between Step 3 and 4. However, the biggest issue is Step 2: it is a no-op step that changes nothing, violating the spirit of 'each step changes one thing' by being a redundant placeholder, or if it's meant to be the expansion, it failed to execute it. Actually, looking closely, Step 2 is literally the same as Step 1. This is a defect. Step 3 then does the expansion. So Step 2 is a redundant no-op. Is a no-op a defect? The prompt says 'Each step changes ONE thing'. A step that changes nothing is arguably a defect in granularity or logic. But let's look at Step 4. Step 4 shows the result of dividing by x^3. Step 3 is the expanded polynomial. The transition 3->4 is 'divide by highest power'. The label 'algebra' is acceptable. Step 5 simplifies 8/16 to 1/2. Label 'algebra' is acceptable. Step 6 takes the limit of the variable parts. Label 'limit' is acceptable. Step 7 is the final value. Let's re-read the contract. 'Each step changes ONE thing.' Step 2 changes nothing. It is identical to Step 1. This is a defect. It adds no value and violates the implicit requirement that steps progress the solution. Also, Step 2's note says 'Expand the cubic terms', but it didn't. Step 3 expands them. So Step 2 is a lie in its note and a no-op in its math. This is an error/misleading.
Senior review claude-sonnet-5-5, 2026-10-06: fail — Line 3 is false as an expression. Expanding gives numerator 8x^3+12x^2+4x and denominator 16x^3+24x^2+12x+7, but the line has -6x and 16x. The 'proved' marks on lines 2-7 appear to compare Limit objects, which simplify evaluates to 1/2 on both sides, so the error was not caught. The final answer 1/2 is correct, but the intermediate algebra in lines 3-5 is wrong. Line 2 is also a no-op whose note claims an expansion.
qwen3.6:27b-mlx: dismiss — Line 2 is a harmless no-op restatement (equal to line 1) and not one of the rule violations listed; the objection also wrongly treats 'simplify' as an invalid label for expanding in line 3, and it missed the real error in line 3.
Every verdict on record (5)
gpt-oss:20b: fail (error) 2026-10-06 — Step 4 incorrectly rewrites the fraction by factoring out 8*x**3/16*x**3 and then mis‑expressing the remaining terms. The correct division by x**3 should yield (8+12/x-6/x**2)/(16+24/x+16/x**2+7/x**3).qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 2 is labeled 'simplify' but the note claims to expand terms, while the expression in Step 2 is identical to Step 1 (no expansion occurred). Step 3 performs the expansion, but the label 'simplify' is vague for an expansion step, and the transition from Step 2 to Step 3 applies the expansion rule without a proper label like 'algebra' or 'expand' (though 'simplify' is in the vocab, the previous step was a no-op). More critically, Step 4 applies 'algebra' to divide by x^3, but the expression shown is already divided and factored, implying the division happened between Step 3 and 4. However, the biggest issue is Step 2: it is a no-op step that changes nothing, violating the spirit of 'each step changes one thing' by being a redundant placeholder, or if it's meant to be the expansion, it failed to execute it. Actually, looking closely, Step 2 is literally the same as Step 1. This is a defect. Step 3 then does the expansion. So Step 2 is a redundant no-op. Is a no-op a defect? The prompt says 'Each step changes ONE thing'. A step that changes nothing is arguably a defect in granularity or logic. But let's look at Step 4. Step 4 shows the result of dividing by x^3. Step 3 is the expanded polynomial. The transition 3->4 is 'divide by highest power'. The label 'algebra' is acceptable. Step 5 simplifies 8/16 to 1/2. Label 'algebra' is acceptable. Step 6 takes the limit of the variable parts. Label 'limit' is acceptable. Step 7 is the final value. Let's re-read the contract. 'Each step changes ONE thing.' Step 2 changes nothing. It is identical to Step 1. This is a defect. It adds no value and violates the implicit requirement that steps progress the solution. Also, Step 2's note says 'Expand the cubic terms', but it didn't. Step 3 expands them. So Step 2 is a lie in its note and a no-op in its math. This is an error/misleading.claude-sonnet-5-5: fail (error) 2026-10-06 — Line 3 is false as an expression. Expanding gives numerator 8x^3+12x^2+4x and denominator 16x^3+24x^2+12x+7, but the line has -6x and 16x. The 'proved' marks on lines 2-7 appear to compare Limit objects, which simplify evaluates to 1/2 on both sides, so the error was not caught. The final answer 1/2 is correct, but the intermediate algebra in lines 3-5 is wrong. Line 2 is also a no-op whose note claims an expansion.qwen3.6:27b-mlx: fail (error) 2026-10-06 — Step 2 claims to expand the cubic terms but leaves the expression in factored form, performing no algebraic change. Step 3 then performs the expansion, but the label 'simplify' is insufficient for expanding a polynomial; more critically, the transition from Step 2 to Step 3 applies the expansion rule, but Step 2 itself is a null step that violates the 'changes ONE thing' rule by doing nothing while claiming to simplify.gpt-oss:20b: pass 2026-10-06
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 2026-10-06 with SymPy 1.14.0.