Limits using Taylor series
Problem 7.709 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 2 x + 2 \ln{\left(x + 1 \right)}}{x^{2}} \).
- Near 0, -x + log(x + 1) = x**3/3 - x**2/2 + ⋯ (its Maclaurin series).
- \[ \lim_{x \to 0}\left(\frac{- \frac{x^{3}}{3} + \frac{x^{2}}{2} - x + \ln{\left(x + 1 \right)}}{x^{3}}\right) = 0 \]The polynomial matches the numerator to that order.✓ Proved
- \[ \frac{\frac{2 x^{3}}{3} - x^{2}}{x^{2}} = \frac{2 x}{3} - 1 \]Divide through by the denominator.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 2 x + 2 \ln{\left(x + 1 \right)}}{x^{2}}\right) = -1 \]Let x → 0: only the constant term survives.✓ Proved
Answer \( -1 \)
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. How we verify
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 | the expression at x = ±10⁻⁸ in 50-digit arithmetic |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 1 contains a sign error in the Maclaurin series expansion: it claims -x + log(x+1) = x^3/3 - x^2/2 + ..., but the correct expansion is -x^2/2 +
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Step 1 contains a sign error in the Maclaurin series expansion: it claims -x + log(x+1) = x^3/3 - x^2/2 + ..., but the correct expansion is -x^2/2 +gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: fail (error) 2026-10-16 — The Maclaurin series expansion in step 1 is incorrect; the term $-x^2/2$ should be $+x^2/2$. Consequently, the numerator is $O(x^3)$, not $O(x^2)$, so the limit is 0, not -1.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution contains a critical algebraic error in Step 3: it divides the numerator term $-x^2/2$ by $x^2$ to get $-1/2$, but then incorrectly writgpt-oss:20b: fail (error) 2026-10-16 — Step 2 presents an incorrect limit expression that does not correspond to the numerator of the problem; it is unnecessary and misleading. The correct series expansion and subsequent steps are fine, but the erroneous equation invalidates the solution’s logical flow.
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/taylor_limit, checked 2026-10-19 with SymPy 1.14.0.