Limit of \( \displaystyle \frac{\sqrt{3 x + 4} - 2}{3 x} \) as \( x \to 0 \)
Problem 1.53 · medium
- \[ \lim_{x \to 0^+}\left(\frac{\sqrt{3 x + 4} - 2}{3 x}\right) \]limitStart with the limit of the original expression.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sqrt{3 x + 4} - 2\right)}{\frac{d}{d x} 3 x}\right) \]lhopitalApply L'Hôpital's rule because the limit is of the form 0/0.✓ Proved
- \[ = \lim_{x \to 0^+}\left(\frac{1}{2 \sqrt{3 x + 4}}\right) \]simplify algebraCalculate the derivatives of the numerator and denominator. Simplify the fraction.✓ Proved
- \[ = \lim_{x \to 0^+} \frac{1}{4} \]simplifyEvaluate the expression inside the limit at x = 0.✓ Proved
- \[ = \frac{1}{4} \]simplifyFinal result.✓ 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 x = 0 undefined where Derivative(3*x, x) = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(3*x, x) = 0 undefined where 3*x + 4 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 4 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x + 4 = 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: passqwen3.6:27b-mlx: pass
Senior review claude-sonnet-5-5, 2026-10-03: pass — L'Hôpital applies to the 0/0 form. Derivative of sqrt(3x+4) is 3/(2 sqrt(3x+4)) and of 3x is 3, giving 1/(2 sqrt(4)) = 1/4. All lines were proved equal by SymPy and the final answer matches the stated one.
qwen3.6:27b-mlx: dismiss — Step 2 leaves unevaluated Derivative objects inside the limit, so step 3 only evaluates them (derivative of 3x+4 inner function included), which counts as simplification; 'simplify' is a legitimate label, and the vocabulary has no derivative rule. Evaluating the derivatives is not a second independent change, and SymPy proved equality.
Every verdict on record (5)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03claude-sonnet-5-5: pass 2026-10-03 — L'Hôpital applies to the 0/0 form. Derivative of sqrt(3x+4) is 3/(2 sqrt(3x+4)) and of 3x is 3, giving 1/(2 sqrt(4)) = 1/4. All lines were proved equal by SymPy and the final answer matches the stated one.qwen3.6:27b-mlx: fail (error) 2026-09-30 — Step 3 applies the differentiation rule (derivative) to compute the derivatives, but is labeled 'simplify'. The label 'simplify' describes algebraic reduction, not differentiation. Additionally, Step 2 claims the form is 0/0, which is correct, but the transition to Step 3 combines the application of L'Hôpital's rule (replacing limit of ratio with limit of ratio of derivatives) and the actual calculation of those derivatives into a single step, violating the 'one rule per step' constraint if we consider the derivative calculation distinct from the rule application. However, the primary defect is the mislabeling of the differentiation step as 'simplify'.gpt-oss:20b: pass 2026-09-30
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 2026-10-03 with SymPy 1.14.0.