Limit of \( \displaystyle \frac{6 x + 3}{\left(3 x + 1\right)^{2} + 3} \) as \( x \to \infty \)
Problem 1.559 · medium
Evaluate \( \displaystyle \lim_{x \to \infty} \frac{6 x + 3}{\left(3 x + 1\right)^{2} + 3} \).
- \[ \lim_{x \to \infty}\left(\frac{6 x + 3}{\left(3 x + 1\right)^{2} + 3}\right) \]limitStart with the limit of the given function.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{6 x + 3}{9 x^{2} + 6 x + 4}\right) \]simplifyExpand the denominator.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{\frac{6}{x} + \frac{3}{x^{2}}}{9 + \frac{6}{x} + \frac{4}{x^{2}}}\right) \]rewrite simplifyDivide numerator and denominator by x^2 to prepare for the limit at infinity. Simplify the terms inside the limit.✓ Proved
- \[ = \lim_{x \to \infty}\left(\frac{6}{x}\right) \left(\lim_{x \to \infty}\left(9 + \frac{6}{x} + \frac{4}{x^{2}}\right)\right)^{-1} \]limit-lawApply the quotient rule for limits.✓ Proved
- \[ = 0 \]limit simplifyEvaluate the limits of the numerator and denominator separately. Final result.✓ Proved
Answer \( 0 \)
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 (3*x + 1)**2 + 3 = 0 undefined where 9*x**2 + 6*x + 4 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 9*x**2 + 6*x + 4 = 0 undefined where x = 0 undefined where 9 + 6/x + 4/x**2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 9 + 6/x + 4/x**2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where 9 + 6/x + 4/x**2 = 0 undefined where Limit(9 + 6/x + 4/x**2, x, oo, dir='-') = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 undefined where Limit(9 + 6/x + 4/x**2, 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: fail (error) — Step 5 incorrectly applies the quotient rule for limits by splitting the numerator into separate limits (Limit(6/x) / Limit(denominator)) instead of keeping the numerator as a single limit (Limit(6/x + 3/x^2)). This is mathematically invalid because the limit of a sum is not the sum of the limits in this context without proper grouping, and specifically, the step implies Limit(A+B)/Limit(C) = Limit(A)/Limit(C) which is not a standard limit law application shown here; it drops the 3/x^2 term from the numerator entirely in the transition to step 6.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 5 incorrectly applies the quotient rule for limits by splitting the numerator into separate limits (Limit(6/x) / Limit(denominator)) instead of keeping the numerator as a single limit (Limit(6/x + 3/x^2)). This is mathematically invalid because the limit of a sum is not the sum of the limits in this context without proper grouping, and specifically, the step implies Limit(A+B)/Limit(C) = Limit(A)/Limit(C) which is not a standard limit law application shown here; it drops the 3/x^2 term from the numerator entirely in the transition to step 6.qwen3.6:27b-mlx: fail (error) 2026-10-12 — Step 5 incorrectly applies the quotient rule for limits by splitting the numerator into separate limits (Limit(6/x) / Limit(denominator)) instead of keeping the numerator as a single limit (Limit(6/x + 3/x^2)). This is algebraically invalid because Limit(A + B) is not equal to Limit(A) in general, and the quotient rule requires Limit(Numerator)/Limit(Denominator), not Limit(Term1)/Limit(Denominator).gpt-oss:20b: fail (error) 2026-10-12 — Step 5 incorrectly applies the limit-law: it splits the numerator limit into only 6/x, dropping the 3/x^2 term. The correct numerator limit is Limit(6/x + 3/x^2, x, oo), not just Limit(6/x, x, oo).
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 2026-10-12 with SymPy 1.14.0.