Home›Calculus 2›Limits using Taylor series›Problem 7.669

Limits using Taylor series

Problem 7.669 · medium

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

Near 0, exp(x**2) - 1 = x**4/2 + x**2 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{4}}{2} - x^{2} + e^{x^{2}} - 1}{x^{4}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{x^{4}}{2} - x^{2}}{x^{2}} = - \frac{x^{2}}{2} - 1 \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{1 - e^{x^{2}}}{x^{2}}\right) = -1 \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( -1 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the Maclaurin series for e^(x^2) and uses it to evaluate the limit. The intermediate steps are algebraically sound and logically consistent.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the Maclaurin series for e^(x^2) and uses it to evaluate the limit. The intermediate steps are algebraically sound and logically consistent.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — Line 1 contains a factual error in the Maclaurin series expansion: it claims exp(x^2) - 1 = x^4/2 + x^2 + ..., but the correct expansion is x^2 + x^4/2 + ... (the x^2 term is missing or misordered in the description, though the subsequent algebra in line 3 uses the correct terms, the initial statement is mathematically incorrect as written). Furthermore, the problem asks for the limit of (1 - e^{x^2})/x^2, but line 2 analyzes a different expression involving x^4 in the denominator, which is irrelevant to the stated problem and suggests a confusion in the solution strategy.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Line 1 states the expansion for exp(x^2) - 1, but the problem requires 1 - exp(x^2). The solution implicitly flips the sign in Line 3 without stating it, creating a logical gap. Additionally, Line 2 is irrelevant to the final limit calculation and confuses the order of terms.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.