Recognising a limit as a derivative
Problem 2.1556 · medium
The limit \( \displaystyle \lim_{h\to 0} \frac{1 - \cos{\left(h \right)}}{h} \) 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) = cos(x) and a = pi.
- \[ \left. \cos{\left(x \right)} \right|_{\substack{ x=\pi }} = -1 \]This is the term being subtracted, f(a).✓ Proved
- \[ \frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)} \]So the limit is f′(a).✓ Proved
- \[ \left. - \sin{\left(x \right)} \right|_{\substack{ x=\pi }} = 0 \]Evaluate at a.✓ Proved
Answer \( f(x) = \cos{\left(x \right)},\ a = \pi;\ \text{the limit is } f'(\pi) = 0 \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: fail (error) — The limit expression is (1 - cos(h))/h, which corresponds to f(x) = cos(x) and a = 0, since f(0) = 1. The solution incorrectly identifies a = pi, where f(pi) = -1, leading to a mismatch in the numerator.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The limit expression is (1 - cos(h))/h, which corresponds to f(x) = cos(x) and a = 0, since f(0) = 1. The solution incorrectly identifies a = pi, where f(pi) = -1, leading to a mismatch in the numerator.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims the limit is the derivative of cos(x) at a = pi, but the numerator is 1 - cos(h), which corresponds to cos(0) - cos(h). This is the derivative of cos(x) at a = 0, not pi. The setup incorrectly identifies 'a'.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.