Limit of \( \displaystyle \frac{\sqrt{x} - 1}{x - 1} \) as \( x \to 1 \)
Problem 1.236 · medium
Evaluate \( \displaystyle \lim_{x \to 1} \frac{\sqrt{x} - 1}{x - 1} \).
- \[ \lim_{x \to 1^+}\left(\frac{\sqrt{x} - 1}{x - 1}\right) \]limit algebraDefine the limit to be evaluated. Multiply the numerator and denominator by the conjugate (sqrt(x) + 1).✓ Proved
- \[ = \lim_{x \to 1^+} \frac{1}{\sqrt{x} + 1} \]simplify simplifySimplify the numerator using the difference of squares. Cancel the common factor (x - 1).✓ Proved
- \[ = \lim_{x \to 1^+} \frac{1}{2} \]limitEvaluate the limit by substituting x = 1.✓ Proved
- \[ = \frac{1}{2} \]limitFinal result.✓ Proved
Answer \( \frac{1}{2} \)
✓ 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 - 1 = 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where x - 1 = 0 undefined where sqrt(x) + 1 = 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(x) + 1 = 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(a - b) reduced to 0 undefined where sqrt(x) + 1 = 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 — The solution correctly applies algebraic manipulation (rationalizing the numerator) and simplification to resolve the indeterminate form, followed by direct substitution. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies algebraic manipulation (rationalizing the numerator) and simplification to resolve the indeterminate form, followed by direct substitution. Each step changes only one aspect of the expression and uses valid labels from the vocabulary.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies algebraic manipulation (rationalization) and simplification rules. Each step changes only one aspect of the expression, and the labels used are appropriate for the operations performed.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.