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Limits using Taylor series

Problem 7.609 · medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{x - e^{x} + 1}{x^{2}} \).

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
\[ \frac{- \frac{x^{3}}{6} - \frac{x^{2}}{2}}{x^{2}} = - \frac{x}{6} - \frac{1}{2} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{x - e^{x} + 1}{x^{2}}\right) = - \frac{1}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( - \frac{1}{2} \)

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: fail (error) — Step 1 contains a sign error in the Maclaurin series expansion: it claims -x + e^x - 1 equals x^2/2 + x^3/6, but the correct expansion is x^2/2 - x^3/6 + ... (since e^x = 1 + x + x^2/2 + x^3/6 + ...). Although the final limit is correct because the x^3 term vanishes upon division by x^2, the intermediate algebraic statement is false.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 1 contains a sign error in the Maclaurin series expansion: it claims -x + e^x - 1 equals x^2/2 + x^3/6, but the correct expansion is x^2/2 - x^3/6 + ... (since e^x = 1 + x + x^2/2 + x^3/6 + ...). Although the final limit is correct because the x^3 term vanishes upon division by x^2, the intermediate algebraic statement is false.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly computes the limit of a different expression (divided by x^3) in step 2, and the sign of the numerator in step 1 is inconsistent with the problem statement, leading to a confusing derivation despite the correct final answer.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution contains a sign error in the setup: it expands -x + e^x - 1 instead of the problem's numerator x - e^x + 1. Although the final numerical answer is correct by coincidence (since the leading term is -x^2/2 in both cases), the intermediate reasoning is flawed and would teach a student to ignore signs.
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.