Recognising a limit as a derivative
Problem 2.1715 · medium
The limit \( \displaystyle \lim_{h\to 0} \frac{\frac{1}{2} + \frac{1}{h - 2}}{h} \) 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) = 1/x and a = -2.
- \[ \left. \frac{1}{x} \right|_{\substack{ x=-2 }} = - \frac{1}{2} \]This is the term being subtracted, f(a).✓ Proved
- \[ \frac{d}{d x} \frac{1}{x} = - \frac{1}{x^{2}} \]So the limit is f′(a).✓ Proved
- \[ \left. - \frac{1}{x^{2}} \right|_{\substack{ x=-2 }} = - \frac{1}{4} \]Evaluate at a.✓ Proved
Answer \( f(x) = \frac{1}{x},\ a = -2;\ \text{the limit is } f'(-2) = - \frac{1}{4} \)
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 setup in line 1 is incorrect. For f(x) = 1/x and a = -2, the term f(a) is -1/2, but the numerator contains +1/2. The limit expression corresponds to f(x) = -1/x with a = -2, or f(x) = 1/x with a = 2 (where f(2)=1/2). As written, the signs do not match the definition of the derivative.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The setup in line 1 is incorrect. For f(x) = 1/x and a = -2, the term f(a) is -1/2, but the numerator contains +1/2. The limit expression corresponds to f(x) = -1/x with a = -2, or f(x) = 1/x with a = 2 (where f(2)=1/2). As written, the signs do not match the definition of the derivative.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution incorrectly identifies a = -2. For f(x) = 1/x, f(-2) = -1/2, but the limit expression has +1/2 in the numerator, which corresponds to f(2) = 1/2. Thus, a must be 2, not -2. The final value -1/4 is coincidentally correct because f'(2) = -1/4 and f'(-2) = -1/4, but the identification of 'a' is wrong.
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.