Recognising a limit as a derivative
Problem 2.1867 · medium
The limit \( \displaystyle \lim_{h\to 0} \frac{\cos{\left(h \right)} - 1}{h} \) 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) = sin(x) and a = pi/2.Reviewed
- \[ \left. \sin{\left(x \right)} \right|_{\substack{ x=\frac{\pi}{2} }} = 1 \]This is the term being subtracted, f(a).✓ Proved
- \[ \frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)} \]So the limit is f′(a).✓ Proved
- \[ \left. \cos{\left(x \right)} \right|_{\substack{ x=\frac{\pi}{2} }} = 0 \]Evaluate at a.✓ Proved
Answer \( f(x) = \sin{\left(x \right)},\ a = \frac{\pi}{2};\ \text{the limit is } f'(\frac{\pi}{2}) = 0 \)
✓ 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 a valid function f(x) = sin(x) and point a = pi/2 such that the given limit represents the derivative f'(a). The algebraic verification steps are correct and the final evaluation is accurate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies a valid function f(x) = sin(x) and point a = pi/2 such that the given limit represents the derivative f'(a). The algebraic verification steps are correct and the final evaluation is accurate.gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly identifies that the given limit matches the definition of the derivative of f(x) = sin(x) at a = pi/2, since f(pi/2) = 1 and f'(x) = cos(x). The evaluation is correct.
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 2026-10-08 with SymPy 1.14.0.