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Limits using Taylor series

Problem 7.707 · medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 x - 3 e^{x} + 3}{x^{2}} \).

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

Answer \( - \frac{3}{2} \)

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) — The solution incorrectly states the Maclaurin series for -x + exp(x) - 1 in step 1, claiming it equals x**3/6 + x**2/2 + ..., whereas the correct expansion is x**2/2 + x**3/6 + ... (the order of terms is swapped, but more importantly, step 2 uses a limit of order 3 which is unnecessary and confusing for a denominator of x**2). Furthermore, step 2 evaluates a limit that is not directly used to derive the final answer in the subsequent steps; the logic jumps from an incorrect series statement to a correct algebraic simplification without properly justifying the truncation of the series for the specific limit required.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — The solution incorrectly states the Maclaurin series for -x + exp(x) - 1 in step 1, claiming it equals x**3/6 + x**2/2 + ..., whereas the correct expansion is x**2/2 + x**3/6 + ... (the order of terms is swapped, but more importantly, step 2 uses a limit of order 3 which is unnecessary and confusing for a denominator of x**2). Furthermore, step 2 evaluates a limit that is not directly used to derive the final answer in the subsequent steps; the logic jumps from an incorrect series statement to a correct algebraic simplification without properly justifying the truncation of the series for the specific limit required.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly factors out -3 from the numerator (3x - 3e^x + 3) to match the series for -x + e^x - 1, but the correct factorization yields -3(-x + e^x - 1), not 3(-x + e^x - 1). Consequently, the sign of the final answer is flipped; the correct limit is 3/2, not -3/2.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly states the Maclaurin series for -x + e^x - 1 as starting with x^3/6 + x^2/2, omitting the crucial x term. This leads to an incorrect numerator in step 2 and a flawed derivation, although the final answer happens to be correct by coincidence.
gpt-oss:20b: pass 2026-10-16

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.