Limit of \( \displaystyle \frac{\sqrt{\left(3 x - 1\right)^{2} + 9} - 3}{\left(3 x - 1\right)^{2}} \) as \( x \to \frac{1}{3} \)
Problem 1.40 · hard Video
Evaluate \( \displaystyle \lim_{x \to \frac{1}{3}} \frac{\sqrt{\left(3 x - 1\right)^{2} + 9} - 3}{\left(3 x - 1\right)^{2}} \).
- \[ \lim_{x \to \frac{1}{3}^+}\left(\frac{\sqrt{\left(3 x - 1\right)^{2} + 9} - 3}{\left(3 x - 1\right)^{2}}\right) \]limit rationalizeSet up the limit with the correct direction. Multiply numerator and denominator by the conjugate.✓ Proved
- \[ = \lim_{x \to \frac{1}{3}^+} \frac{1}{\sqrt{\left(3 x - 1\right)^{2} + 9} + 3} \]algebra simplify cancelExpand the squared term in the numerator. Simplify the numerator. Cancel the common factor (3*x - 1)**2.✓ Proved
- \[ = \frac{1}{6} \]limit algebra simplify simplifyEvaluate the limit by substituting x = 1/3. Simplify the expression inside the square root. Simplify the square root and the sum. Final result.✓ Proved
Answer \( \frac{1}{6} \)
✓ 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. 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 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where 3*x - 1 = 0 undefined where sqrt((3*x - 1)**2 + 9) + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt((3*x - 1)**2 + 9) + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt((3*x - 1)**2 + 9) + 3 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt((3*x - 1)**2 + 9) + 3 = 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 |
| 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 the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses valid labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the conjugate method to resolve the indeterminate form. Each step changes only one aspect of the expression and uses appropriate labels from the fixed vocabulary.gpt-oss:20b: pass 2026-09-29
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.