Limit of \( \displaystyle \frac{- 3 x + \left(3 x + 1\right)^{3} - 1}{2 \left(3 x + 1\right)^{3} + 5} \) as \( x \to \infty \)
Problem 1.417 · hard
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{- 3 x + \left(3 x + 1\right)^{3} - 1}{2 \left(3 x + 1\right)^{3} + 5} \).
- \[ \lim_{x \to \infty}\left(\frac{- 3 x + \left(3 x + 1\right)^{3} - 1}{2 \left(3 x + 1\right)^{3} + 5}\right) \]limitEvaluate the limit as x approaches infinity.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{27 x^{3} + 27 x^{2} + 6 x}{54 x^{3} + 54 x^{2} + 18 x + 7}\right) \]simplify simplifyExpand the cubic term (3x+1)**3. Combine like terms in the numerator and denominator.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{27 + \frac{27}{x} + \frac{6}{x^{2}}}{54 + \frac{54}{x} + \frac{18}{x^{2}} + \frac{7}{x^{3}}}\right) \]algebra simplifyDivide both numerator and denominator by the highest power of x, which is x**3. Simplify the fractions by dividing each term by its power of x.✓ Proved
- \[ = \lim_{x \to \infty} \frac{1}{2} \]limit-lawApply the limit laws as terms with 1/x^n approach zero.✓ Proved
- \[ = \frac{1}{2} \]simplifyReduce the fraction to its simplest form.✓ 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. The reviewers disagree about how one step is explained; every verdict is in 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*(3*x + 1)**3 + 5 = 0 undefined where 54*x**3 + 54*x**2 + 18*x + 7 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 54*x**3 + 54*x**2 + 18*x + 7 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 54*x**3 + 54*x**2 + 18*x + 7 = 0 undefined where x = 0 undefined where 54 + 54/x + 18/x**2 + 7/x**3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 54 + 54/x + 18/x**2 + 7/x**3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 54 + 54/x + 18/x**2 + 7/x**3 = 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 3 contains multiple errors: the coefficient 2_7 is a typo, the numerator should be 27*x**3+27*x**2+6*x (the 27*x term cancels), and the denominator should be 54*x**3+54*x**2+18*x+5, not 7. This step applies more than one rule at once and changes several terms simultaneously, violating the contract.qwen3.6:27b-mlx: pass — The solution correctly expands, simplifies, and evaluates the limit by dividing by the highest power of x. Each step applies a single rule and is labeled correctly.
Every verdict on record (4)
gpt-oss:20b: fail (error) 2026-10-08 — Step 3 contains multiple errors: the coefficient 2_7 is a typo, the numerator should be 27*x**3+27*x**2+6*x (the 27*x term cancels), and the denominator should be 54*x**3+54*x**2+18*x+5, not 7. This step applies more than one rule at once and changes several terms simultaneously, violating the contract.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly expands, simplifies, and evaluates the limit by dividing by the highest power of x. Each step applies a single rule and is labeled correctly.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly expands, simplifies, and evaluates the limit using standard algebraic techniques. Each step applies a single rule and is correctly labeled.gpt-oss:20b: pass 2026-10-08
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.