Limit of \( \displaystyle \frac{\sqrt{3 x + 3} - 2}{3 x - 1} \) as \( x \to \frac{1}{3} \)
Problem 1.106 · medium
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sqrt{3 x + 3} - 2}{3 x - 1} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 3} - 2}{3 x - 1}\right) \]limit algebraSet up the limit with the specified direction. Multiply the numerator and denominator by the conjugate.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 3} + 2} \]simplify simplify cancelExpand the numerator. Simplify the numerator. Cancel the common factor (3*x - 1).✓ Proved
- \[ = \frac{1}{4} \]limit simplify simplify simplify simplifyEvaluate the limit by substituting x = 1/3. Simplify the expression inside the square root. Calculate the square root. Simplify the denominator. Final result.✓ Proved
Answer \( \frac{1}{4} \)
✓ 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 3*x - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 undefined where sqrt(3*x + 3) + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(3*x + 3) + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(3*x + 3) + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(3*x + 3) + 2 = 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step isolates a single algebraic operation or limit evaluation, and the labels accurately reflect the transformations performed.gpt-oss:20b: pass 2026-10-03
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.