Limit of \( \displaystyle \frac{\sqrt{3 x + 5} - 2}{3 x + 1} \) as \( x \to - \frac{1}{3} \)
Problem 1.35 · medium Video
Evaluate \( \displaystyle \lim_{x \to - \frac{1}{3}} \frac{\sqrt{3 x + 5} - 2}{3 x + 1} \).
- \[ \lim_{x \to - \frac{1}{3}^+}\left(\frac{\sqrt{3 x + 5} - 2}{3 x + 1}\right) \]limit algebraSet up the limit from the right. Multiply the numerator and denominator by the conjugate.✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{\sqrt{3 x + 5} + 2} \]simplify algebra cancelSimplify the numerator using the difference of squares. Group the terms to prepare for cancellation. Cancel the common factor (3*x + 1).✓ Proved
- \[ = \lim_{x \to - \frac{1}{3}^+} \frac{1}{4} \]limit simplify simplify simplifyEvaluate the limit by substituting x = -1/3. Simplify the expression inside the square root. Calculate the square root. Add the numbers in the denominator.✓ Proved
- \[ = \frac{1}{4} \]simplifyFinal 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. 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 + 5) + 2 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(3*x + 5) + 2 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(3*x + 5) + 2 = 0 |
| 6 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(3*x + 5) + 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 (6)
qwen3.6:27b-mlx: pass 2026-10-03gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic manipulation (conjugate multiplication) to resolve the indeterminate form, followed by cancellation and direct substitution. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies algebraic manipulation (rationalization) to resolve the indeterminate form, followed by valid simplification and substitution steps. 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.