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Limits using Taylor series

Problem 7.605 · medium

Use Taylor series to evaluate $ \displaystyle \lim_{x\to 0} \frac{2 x - 2 e^{x} + 2}{x^{2}} $.

Near 0, -x + exp(x) - 1 = x**3/6 + x**2/2 + ⋯ (its Maclaurin series).
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{3}}{6} - \frac{x^{2}}{2} - x + e^{x} - 1}{x^{3}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{x^{3}}{3} - x^{2}}{x^{2}} = - \frac{x}{3} - 1 \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{2 x - 2 e^{x} + 2}{x^{2}}\right) = -1 \]
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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution contains a sign error in the setup. The numerator is $2x - 2e^x + 2 = -2(e^x - x - 1)$. Since $e^x - x - 1 = x^2/2 + x^3/6 + \\dots$, t

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution contains a sign error in the setup. The numerator is $2x - 2e^x + 2 = -2(e^x - x - 1)$. Since $e^x - x - 1 = x^2/2 + x^3/6 + \\dots$, t
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly analyzes the limit of a different expression in step 2 (dividing by x^3 instead of x^2), and the logic in step 4 is flawed because it claims the limit is -1 based on the constant term of the expansion from step 3, ignoring that step 2 was irrelevant and the connection between the numerator's expansion and the final result is not properly justified in the text.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly substitutes the expansion for `-x + exp(x) - 1` into the numerator `2x - 2exp(x) + 2`. The numerator is equal to `-2(-x + exp(x) - 1)`, so the substitution should yield `-2(x^2/2 + x^3/6 + ...)`, not `+2(x^2/2 + x^3/6 + ...)`. This sign error leads to the incorrect final answer of -1 instead of the correct answer 1.
gpt-oss:20b: pass 2026-10-14

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 2026-10-17 with SymPy 1.14.0.