Limits using Taylor series
Problem 7.600 · 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: passqwen3.6:27b-mlx: fail (error) — Step 1 claims the Maclaurin series for -x + exp(x) - 1 starts with x^3/6 + x^2/2, but the x term is missing; the expansion is actually x^2/2 + x^3/6 + ... (the x terms cancel). While the final limit is correct, the intermediate series representation in Step 1 is mathematically incorrect as written (it implies the coefficient of x is 0 without showing the cancellation, and lists terms out of order or omits the leading non-zero term's context properly). More critically, Step 2 uses a limit with denominator x^3 to justify the series, but the problem has denominator x^2. The logic is disjointed: Step 1 gives a series, Step 2 checks a different limit, Step 3 divides the series from Step 1 by x^2. The series in Step 1 is technically wrong because it lists x^3/6 before x^2/2 and omits the explicit cancellation of the linear term which is crucial for the x^2 denominator context. Actually, the biggest error is that Step 1 says '-x + exp(x) - 1 = x**3/6 + x**2/2 + ...'. This is false notation; it should be 'x^2/2 + x^3/6 + ...'. Writing it as x^3/6 + x^2/2 suggests the x^3 term is dominant or comes first, which is false. Furthermore, Step 2 is irrelevant to the direct evaluation in Step 3/4. The solution is confusing and contains a formally incorrect series expansion statement.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 1 claims the Maclaurin series for -x + exp(x) - 1 starts with x^3/6 + x^2/2, but the x term is missing; the expansion is actually x^2/2 + x^3/6 + ... (the x terms cancel). While the final limit is correct, the intermediate series representation in Step 1 is mathematically incorrect as written (it implies the coefficient of x is 0 without showing the cancellation, and lists terms out of order or omits the leading non-zero term's context properly). More critically, Step 2 uses a limit with denominator x^3 to justify the series, but the problem has denominator x^2. The logic is disjointed: Step 1 gives a series, Step 2 checks a different limit, Step 3 divides the series from Step 1 by x^2. The series in Step 1 is technically wrong because it lists x^3/6 before x^2/2 and omits the explicit cancellation of the linear term which is crucial for the x^2 denominator context. Actually, the biggest error is that Step 1 says '-x + exp(x) - 1 = x**3/6 + x**2/2 + ...'. This is false notation; it should be 'x^2/2 + x^3/6 + ...'. Writing it as x^3/6 + x^2/2 suggests the x^3 term is dominant or comes first, which is false. Furthermore, Step 2 is irrelevant to the direct evaluation in Step 3/4. The solution is confusing and contains a formally incorrect series expansion statement.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: fail (error) 2026-10-15 — Line 1 incorrectly states the Maclaurin series for -x + e^x - 1 as x^3/6 + x^2/2 + ..., omitting the linear term (which is actually 0) but more critically, the expansion of e^x is 1 + x + x^2/2 + x^3/6 + ..., so -x + e^x - 1 = x^2/2 + x^3/6 + ... The sentence lists terms in descending order of power (x^3 then x^2) which is non-standard and confusing, but the main error is in Line 2 where it divides by x^3 instead of x^2 as per the problem statement. The problem asks for the limit of the expression divided by x^2, but Line 2 computes a limit divided by x^3. This is a fundamental mismatch with the problem statement.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly expands the numerator as -x + e^x - 1, omitting the factor of 3 present in the problem statement (-3x + 3e^x - 3). Although the final answer is coincidentally correct because the factor of 3 is applied in step 3, the initial setup in step 1 is mathematically inconsistent with the problem statement.gpt-oss:20b: fail (error) 2026-10-14 — The solution misapplies the Taylor expansion: the numerator should be 3(-x+e^x-1)=3(x^2/2+x^3/6+…), but the steps use a wrong sign and an incorrect intermediate expression, leading to an unjustified limit calculation.
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.