Limit of \( \displaystyle \frac{\sqrt{3} \sqrt{x} - 1}{3 x - 1} \) as \( x \to \frac{1}{3} \)
Problem 1.359 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sqrt{3} \sqrt{x} - 1}{3 x - 1} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{3} \sqrt{x} - 1}{3 x - 1}\right) \]limitStart with the limit of the original function.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+} \text{NaN} \]simplifyCheck the indeterminate form.Reviewed
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\frac{d}{d x} \left(\sqrt{3} \sqrt{x} - 1\right)}{\frac{d}{d x} \left(3 x - 1\right)}\right) \]lhopitalApply L'Hôpital's Rule because the limit is of the form 0/0.Reviewed
- \[ = \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{3}}{6 \sqrt{x}}\right) \]simplify algebraCompute the derivatives. Simplify the fraction.✓ Proved
- \[ = \frac{1}{2} \]limit simplify algebra simplify simplifyEvaluate the limit by substituting x = 1/3. Simplify the denominator. Multiply by the reciprocal. Simplify the fraction. Final result.✓ Proved
Answer \( \frac{1}{2} \)
Lines: 9 proved, 2 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | simplify left nan; no point in the sample was defined on both lines undefined where 3*x - 1 = 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | simplify left nan; no point in the sample was defined on both lines undefined where Derivative(3*x - 1, x) = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where Derivative(3*x - 1, x) = 0 undefined where x = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x = 0 |
| 7 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 8 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 9 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 |
| 10 | ✓ 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 — The solution correctly applies L'Hôpital's Rule and simplifies the resulting expression step-by-step. Each step changes only one thing and uses valid labels from the fixed vocabulary.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies L'Hôpital's Rule and simplifies the resulting expression step-by-step. Each step changes only one thing and uses valid labels from the fixed vocabulary.qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly identifies the indeterminate form, applies L'Hôpital's Rule, and simplifies the resulting expression to the correct limit. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.