Limits using Taylor series
Problem 7.683 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{2 e^{x^{2}} - 2}{x^{2}} \).
- 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
- \[ \frac{x^{4} + 2 x^{2}}{x^{2}} = x^{2} + 2 \]Divide through by the denominator.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{2 e^{x^{2}} - 2}{x^{2}}\right) = 2 \]Let x → 0: only the constant term survives.✓ Proved
Answer \( 2 \)
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: passqwen3.6:27b-mlx: fail (error) — Step 1 states an incorrect Maclaurin series for e^(x^2) - 1 (it claims the x^2 term is missing and includes an x^4 term incorrectly). Although the final limit is correct, the derivation relies on a false premise that would teach a student the wrong expansion.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — Step 1 states an incorrect Maclaurin series for e^(x^2) - 1 (it claims the x^2 term is missing and includes an x^4 term incorrectly). Although the final limit is correct, the derivation relies on a false premise that would teach a student the wrong expansion.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Line 1 contains an incorrect Maclaurin series expansion for e^(x^2) - 1; it lists x^4/2 + x^2 instead of the correct x^2 + x^4/2 + ... (though the tgpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The Maclaurin series for exp(x^2) is 1 + x^2 + x^4/2 + ..., so exp(x^2) - 1 = x^2 + x^4/2 + .... The solution incorrectly states the series as x^4/2 + x^2 + ... (implying the x^2 term is missing or misordered in the expansion logic, though the terms are listed, the subsequent algebraic manipulation in step 2 is inconsistent with the problem statement). More critically, Step 2 introduces a limit with denominator x^4 and a numerator involving -x^4/2 - x^2 + exp(x^2) - 1, which evaluates to 0, but this step is irrelevant to the original problem which has denominator x^2. The solution confuses the order of the limit or introduces an unnecessary and confusing intermediate step that does not follow from the previous sentence. The final answer is correct, but the reasoning is flawed and misleading.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.