Recognising a limit as a derivative
Problem 2.1711 · medium
The limit \( \displaystyle \lim_{h\to 0} \frac{\sqrt{h + 25} - 5}{h} \) is the derivative of some function \( \displaystyle f \) at some number \( \displaystyle a \). Find \( \displaystyle f \) and \( \displaystyle a \), and evaluate the limit.
- The limit has the shape [f(a + h) − f(a)]/h with f(x) = sqrt(x) and a = 25.Reviewed
- \[ \left. \sqrt{x} \right|_{\substack{ x=25 }} = 5 \]This is the term being subtracted, f(a).✓ Proved
- \[ \frac{d}{d x} \sqrt{x} = \frac{1}{2 \sqrt{x}} \]So the limit is f′(a).✓ Proved
- \[ \left. \frac{1}{2 \sqrt{x}} \right|_{\substack{ x=25 }} = \frac{1}{10} \]Evaluate at a.✓ Proved
Answer \( f(x) = \sqrt{x},\ a = 25;\ \text{the limit is } f'(25) = \frac{1}{10} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy takes the limit in h directly, without recognising a derivative |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the function and point by matching the limit to the definition of the derivative. The algebraic steps and final evaluation are correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the function and point by matching the limit to the definition of the derivative. The algebraic steps and final evaluation are correct.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the function and point from the limit definition of the derivative and evaluates it accurately.
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
generator:structured/limit_as_derivative, checked 2026-10-06 with SymPy 1.14.0.