Recognising a limit as a derivative
Problem 2.1562 · medium
The limit \( \displaystyle \lim_{h\to 0} \frac{\cos{\left(h + \frac{\pi}{4} \right)} - \frac{\sqrt{2}}{2}}{h} \) 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/4.Reviewed
- \[ \left. \cos{\left(x \right)} \right|_{\substack{ x=\frac{\pi}{4} }} = \frac{\sqrt{2}}{2} \]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
- \[ \left. - \sin{\left(x \right)} \right|_{\substack{ x=\frac{\pi}{4} }} = - \frac{\sqrt{2}}{2} \]Evaluate at a.✓ Proved
Answer \( f(x) = \cos{\left(x \right)},\ a = \frac{\pi}{4};\ \text{the limit is } f'(\frac{\pi}{4}) = - \frac{\sqrt{2}}{2} \)
✓ 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 from the definition of the derivative and evaluates the limit accurately.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the function and point from the definition of the derivative and evaluates the limit accurately.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the function and point from the definition of the derivative and evaluates the limit accurately.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.