Limit of \( \displaystyle \frac{8 x^{3} - 2 x}{16 x^{3} + 5} \) as \( x \to \infty \)
Problem 1.249 · medium
- \[ \lim_{x \to \infty}\left(\frac{8 x^{3} - 2 x}{16 x^{3} + 5}\right) \]limit rewriteStart with the original limit. Divide numerator and denominator by the highest power of x.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{8 - \frac{2}{x^{2}}}{16 + \frac{5}{x^{3}}}\right) \]simplifySimplify the expression inside the limit.✓ Proved
- \[ = \lim_{x \to \infty}\left(8 - \frac{2}{x^{2}}\right) \left(\lim_{x \to \infty}\left(16 + \frac{5}{x^{3}}\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = \lim_{x \to \infty} 8 \left(\lim_{x \to \infty} 16\right)^{-1} \]limit-lawEvaluate the limits of the numerator and denominator separately.✓ Proved
- \[ = \frac{1}{2} \]limit simplifyThe limits evaluate to constant values. Simplify the fraction.✓ Proved
✓ 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 16*x**3 + 5 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 16*x**3 + 5 = 0 undefined where x = 0 undefined where 16 + 5/x**3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 16 + 5/x**3 = 0 undefined where Limit(16 + 5/x**3, x, oo, dir='-') = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where Limit(16 + 5/x**3, x, oo, dir='-') = 0 undefined where Limit(16, x, oo, dir='-') = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Limit(16, x, oo, dir='-') = 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: passqwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-05: fail — The math and the final answer 1/2 are correct. Step 5 compresses the sum law and the evaluation of the vanishing terms (2/x**2 and 5/x**3 tending to 0) into one jump. It should be split into a sum-law step followed by a limit step showing those terms go to 0.
qwen3.6:27b-mlx: uphold — The claim of mathematical incorrectness is overstated, since both sides equal 1/2 and SymPy confirmed it. But step 5 does several independent things at once: the sum law in numerator and denominator, evaluating the limits of 2/x**2 and 5/x**3 to 0, and dropping those terms. It is labelled only 'limit-law', so the justification for removing the vanishing terms is skipped.
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-05gpt-oss:20b: pass 2026-10-05claude-sonnet-5-5: fail (style) 2026-10-05 — The math and the final answer 1/2 are correct. Step 5 compresses the sum law and the evaluation of the vanishing terms (2/x**2 and 5/x**3 tending to 0) into one jump. It should be split into a sum-law step followed by a limit step showing those terms go to 0.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 5 is mathematically incorrect because it implicitly discards the terms -2/x**2 and 5/x**3 without applying a limit law or simplification rule to justify their removal. The transition from Step 4 to Step 5 requires evaluating the limits of the vanishing terms (which is 0), but the step jumps directly to the constants, skipping the necessary logical bridge.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.