Limits using Taylor series
Problem 7.612 · medium
Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{- 3 x + 3 \operatorname{atan}{\left(x \right)}}{x^{3}} \).
- Near 0, x - atan(x) = -x**5/5 + x**3/3 + ⋯ (its Maclaurin series).
- \[ \lim_{x \to 0}\left(\frac{\frac{x^{5}}{5} - \frac{x^{3}}{3} + x - \operatorname{atan}{\left(x \right)}}{x^{5}}\right) = 0 \]The polynomial matches the numerator to that order.✓ Proved
- \[ \frac{\frac{3 x^{5}}{5} - x^{3}}{x^{3}} = \frac{3 x^{2}}{5} - 1 \]Divide through by the denominator.✓ Proved
- \[ \lim_{x \to 0}\left(\frac{- 3 x + 3 \operatorname{atan}{\left(x \right)}}{x^{3}}\right) = -1 \]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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The Maclaurin series for atan(x) is x - x^3/3 + x^5/5 - ..., so x - atan(x) = x^3/3 - x^5/5 + ... The solution incorrectly states the series as -x^5
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The Maclaurin series for atan(x) is x - x^3/3 + x^5/5 - ..., so x - atan(x) = x^3/3 - x^5/5 + ... The solution incorrectly states the series as -x^5gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The Maclaurin series for atan(x) is x - x^3/3 + x^5/5 - ..., so x - atan(x) = x^3/3 - x^5/5 + ... The solution incorrectly states the series as -x^5/gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — Line 1 states the series for x - atan(x) as -x^5/5 + x^3/3, which is incorrect; the correct expansion is x^3/3 - x^5/5 + ... (the signs are swapped in the statement, though the subsequent algebra in Line 3 accidentally uses the correct coefficients for the numerator -3x + 3atan(x) = -3(x - atan(x)). However, Line 1 explicitly claims x - atan(x) equals a series starting with a negative x^5 term, which is false. More critically, Line 1 is presented as a fact about x - atan(x), but the problem involves -3x + 3atan(x) = -3(x - atan(x)). The series for x - atan(x) is x^3/3 - x^5/5 + O(x^7). Line 1 writes it as -x^5/5 + x^3/3, which is numerically correct but conventionally written with increasing powers. The real error is that Line 1 is just a statement, but Line 3 calculates (-3 * (-x^5/5 + x^3/3)) / x^3. If we use the correct series x - atan(x) = x^3/3 - x^5/5, then -3(x - atan(x)) = -x^3 + 3x^5/5. Dividing by x^3 gives -1 + 3x^2/5. The limit is -1. The solution's Line 1 is technically correct in value (-x^5/5 + x^3/3 is the same as x^3/3 - x^5/5), but it's confusingly ordered. Wait, let's look closer. Line 1 says x - atan(x) = -x^5/5 + x^3/3. This is correct. Line 3 computes (-3 * (-x^5/5 + x^3/3)) / x^3. This equals (-3 * (x^3/3 - x^5/5)) / x^3 = (-x^3 + 3x^5/5) / x^3 = -1 + 3x^2/5. The limit is -1. The solution is actually correct in its result. Is there a subtle error? Line 1: x - atan(x) = x - (x - x^3/3 + x^5/5 - ...) = x^3/3 - x^5/5 + ... Line 1 writes -x^5/5 + x^3/3. This is correct. Line 3: Numerator is -3x + 3atan(x) = -3(x - atan(x)). So -3 * (x^3/3 - x^5/5) = -x^3 + 3x^5/5. Line 3 calculates (-3 * (-x^5/5 + x^3/3)) = 3x^5/5 - x^3. Divided by x^3 is 3x^2/5 - 1. Limit is -1. The solution is correct. Why did I think it was wrong? Ah, I need to check if the prompt implies the series in Line 1 is wrong. No, it's just ordered differently. Is there a 'style' issue? Or is it 'pass'? Let's re-read carefully. 'Near 0, x - atan(x) = -x**5/5 + x**3/3 + ...'. This is correct. The subsequent steps follow. The verdict should be pass.gpt-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.