Limit of \( \displaystyle \frac{\sqrt{x} - 3}{x - 9} \) as \( x \to 9 \)
Problem 1.354 · medium
Evaluate \( \displaystyle \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} \).
- \[ \lim_{x \to 9^+}\left(\frac{\sqrt{x} - 3}{x - 9}\right) \]limit algebraSet up the limit from the right side. Multiply the numerator and denominator by the conjugate.✓ Proved
- \[ = \lim_{x \to 9^+} \frac{1}{\sqrt{x} + 3} \]simplify cancelSimplify the numerator using the difference of squares. Cancel the common factor (x - 9).✓ Proved
- \[ = \frac{1}{6} \]limit simplifyEvaluate the limit by substituting x = 9. 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.
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 - 9 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 9 = 0 undefined where sqrt(x) + 3 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(x) + 3 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(x) + 3 = 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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-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.