Limits using Taylor series
Problem 7.610 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{e^{x^{2}} - 1}{x^{2}} \).
- Near 0, exp(x**2) - 1 = x**4/2 + x**2 + ⋯ (its Maclaurin series).
- \[ \lim_{x \to 0}\left(\frac{- \frac{x^{4}}{2} - x^{2} + e^{x^{2}} - 1}{x^{4}}\right) = 0 \]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
- \[ \lim_{x \to 0}\left(\frac{e^{x^{2}} - 1}{x^{2}}\right) = 1 \]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: fail (error) — Line 1 contains a sign error in the Maclaurin series expansion, stating the term is -x^4/2 instead of +x^4/2. Although the final limit is correct, the intermediate algebraic setup is mathematically false.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — Line 1 contains a sign error in the Maclaurin series expansion, stating the term is -x^4/2 instead of +x^4/2. Although the final limit is correct, the intermediate algebraic setup is mathematically false.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Line 1 contains a sign 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^4/2gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Line 1 contains a sign error in the Maclaurin series expansion: it states exp(x**2) - 1 = x**4/2 + x**2, but the correct expansion is x**2 + x**4/2gpt-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.