Home›Calculus 1›Limits at infinity›Problem 1.452

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 1.452 · hard

Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- 2 x + \left(2 x - 1\right)^{3} + 1}{2 \left(2 x - 1\right)^{3} + 5} \).

\[ \lim_{x \to \infty}\left(\frac{- 2 x + \left(2 x - 1\right)^{3} + 1}{2 \left(2 x - 1\right)^{3} + 5}\right) \]
limitStart with the limit of the given function.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{8 x^{3} - 12 x^{2} + 6 x - 2}{16 x^{3} - 24 x^{2} + 8 x + 3}\right) \]
simplifyExpand the cubic terms in the numerator and denominator.✓ Proved
\[ = \lim_{x \to \infty}\left(\frac{8 - \frac{12}{x} + \frac{6}{x^{2}} - \frac{2}{x^{3}}}{16 - \frac{24}{x} + \frac{8}{x^{2}} + \frac{3}{x^{3}}}\right) \]
algebra simplifyDivide both numerator and denominator by the highest power of x, which is x**3. Simplify the fractions within the limit.✓ Proved
\[ = \lim_{x \to \infty} \frac{1}{2} \]
limit-lawApply the limit laws as terms with x in the denominator approach zero.✓ Proved
\[ = 0.5 \]
limitEvaluate the final fraction.✓ 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(2x - 1)*3 + 5 = 0 undefined where 16x*3 - 24x*2 + 8x + 3 = 0
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where 16x3 - 24x*2 + 8x + 3 = 0 undefined where x = 0 undefined where 16 - 24/x + 8/x2 + 3/x3 = 0
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where 16 - 24/x + 8/x2 + 3/x3 = 0
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where x = 0 undefined where 16 - 24/x + 8/x2 + 3/x3 = 0
6	✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly expands, simplifies, and applies limit laws to evaluate the limit. Each step applies a single rule and uses valid labels from the fixed vocabulary.

Senior review claude-sonnet-5-5, 2026-10-09: fail — Line 2 is not equal to line 1: (2x-1)^3 = 8x^3-12x^2+6x-1, so the numerator is 8x^3-12x^2+4x and the denominator is 16x^3-24x^2+12x+3. The later lines inherit the wrong polynomials, though the leading-coefficient ratio 1/2 and the stated answer are still correct.

gpt-oss:20b: uphold — Expanding gives numerator 8x^3-12x^2+4x and denominator 16x^3-24x^2+12x+3, so line 2's 6x-2 and 8x are false. SymPy's 'proved' result likely came from comparing the evaluated limits (both 1/2), not the integrands.

Every verdict on record (5)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly expands, simplifies, and applies limit laws to evaluate the limit. Each step applies a single rule and uses valid labels from the fixed vocabulary.
claude-sonnet-5-5: fail (error) 2026-10-09 — Line 2 is not equal to line 1: (2x-1)^3 = 8x^3-12x^2+6x-1, so the numerator is 8x^3-12x^2+4x and the denominator is 16x^3-24x^2+12x+3. The later lines inherit the wrong polynomials, though the leading-coefficient ratio 1/2 and the stated answer are still correct.
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly expands, simplifies, and applies limit laws to evaluate the limit at infinity. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
gpt-oss:20b: fail (error) 2026-10-09 — Step 2 incorrectly expands the cubic terms: the numerator should be 8*x**3 - 12*x**2 + 4*x, not 8*x**3 - 12*x**2 + 6*x - 2, and the denominator should be 16*x**3 - 24*x**2 + 12*x + 3, not 16*x**3 - 24*x**2 + 8*x + 3. These algebraic errors propagate through subsequent steps.

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 2026-10-09 with SymPy 1.14.0.