Limit of \( \displaystyle \frac{\left(3 x + 1\right)^{3} - 1}{\left(3 x + 1\right)^{2} - 1} \) as \( x \to 0 \)
Problem 1.774 · medium Video
Evaluate \( \displaystyle \lim_{x \to 0} \frac{\left(3 x + 1\right)^{3} - 1}{\left(3 x + 1\right)^{2} - 1} \).
- \[ \lim_{x \to 0^+}\left(\frac{\left(3 x + 1\right)^{3} - 1}{\left(3 x + 1\right)^{2} - 1}\right) \]limitStart with the limit of the original function.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\left(3 x + 1\right)^{3} - 1\right)}{\frac{d}{d x} \left(\left(3 x + 1\right)^{2} - 1\right)}\right) \]lhopitalApply L'Hopital's rule because the limit is an indeterminate form 0/0.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{3 \left(3 x + 1\right)^{2}}{6 x + 2}\right) \]simplifyCompute the derivatives of the numerator and denominator.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{9 x}{2} + \frac{3}{2}\right) \]simplifySimplify the expression by canceling common terms.✓ Proved
- \[ = \frac{3}{2} \]limit simplifyEvaluate the limit by substituting x = 0. Final numerical result.✓ Proved
Answer \( \frac{3}{2} \)
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. How we verify
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 - 1 = 0 undefined where Derivative((3*x + 1)**2 - 1, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative((3*x + 1)**2 - 1, x) = 0 undefined where 6*x + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 6*x + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 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: fail (error) — Step 3 incorrectly differentiates both the numerator and denominator. The derivative of <(3*x+1)^3> is 9*(3*x+1)^2, not 3*(3*x+1)^2, and the derivative of <(3*x+1)^2> is 6*(3*x+1), not 2*(3*x+1). This error propagates through the subsequent steps.qwen3.6:27b-mlx: pass — The solution correctly applies L'Hopital's rule, computes the derivatives, simplifies the resulting expression, and evaluates the limit. Each step changes only one aspect of the expression and uses an acceptable label from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-31 — The solution correctly applies L'Hopital's rule, computes the derivatives, simplifies the resulting expression, and evaluates the limit. Each step changes only one aspect of the expression and uses an acceptable label from the fixed vocabulary.gpt-oss:20b: fail (error) 2026-10-31 — Step 3 incorrectly differentiates both the numerator and denominator. The derivative of <(3*x+1)^3> is 9*(3*x+1)^2, not 3*(3*x+1)^2, and the derivative of <(3*x+1)^2> is 6*(3*x+1), not 2*(3*x+1). This error propagates through the subsequent steps.qwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly applies L'Hopital's rule and simplifies the resulting expression step-by-step. Each step changes only one aspect of the expression and uses an appropriate label from the fixed vocabulary.gpt-oss:20b: inconclusive 2026-10-30 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"Step 3 incorrectly applies the chain rule: the derivative of \((3x+1)^3\) is \(9(3x+1)^2\), not \(3(3x+1)^2\); similarly, the derivative of \((3x+1)^2\) iqwen3.6:27b-mlx: pass 2026-10-30 — The solution correctly applies L'Hopital's rule, computes the derivatives, simplifies the resulting expression, and evaluates the limit. Each step changes one thing and uses an acceptable label.gpt-oss:20b: fail (error) 2026-10-30 — Step 3 incorrectly differentiates the numerator and denominator, missing the factor of 3 from the chain rule. The correct derivatives are 9(3x+1)^2 and 6(3x+1), so the subsequent simplifications are invalid.
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 2026-10-31 with SymPy 1.14.0.