Home›Calculus 2›Limits using Taylor series›Problem 7.607

Limits using Taylor series

Problem 7.607 · medium

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

Near 0, -x + log(x + 1) = x**3/3 - x**2/2 + ⋯ (its Maclaurin series).
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{3}}{3} + \frac{x^{2}}{2} - x + \ln{\left(x + 1 \right)}}{x^{3}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{\frac{x^{3}}{3} - \frac{x^{2}}{2}}{x^{2}} = \frac{x}{3} - \frac{1}{2} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{- x + \ln{\left(x + 1 \right)}}{x^{2}}\right) = - \frac{1}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( - \frac{1}{2} \)

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: pass
qwen3.6:27b-mlx: fail (error) — Step 1 contains a sign error in the Maclaurin series expansion: the term is -x^3/3, not +x^3/3. Consequently, Step 2 uses an incorrect numerator (-x^3/3 instead of +x^3/3) to claim the remainder is O(x^3), which is technically true for the limit but relies on the wrong series coefficient. More importantly, the logic in Step 2 is confusing and potentially misleading regarding the order of the error term relative to the denominator.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — Step 1 contains a sign error in the Maclaurin series expansion: the term is -x^3/3, not +x^3/3. Consequently, Step 2 uses an incorrect numerator (-x^3/3 instead of +x^3/3) to claim the remainder is O(x^3), which is technically true for the limit but relies on the wrong series coefficient. More importantly, the logic in Step 2 is confusing and potentially misleading regarding the order of the error term relative to the denominator.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-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 -x + ln(x+1) = x^3/3 - x^2/2 + ..., but the correct expansion is -x^
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — Line 1 contains a sign error in the Maclaurin series expansion: the term is -x^3/3, not +x^3/3. Although the final answer is correct because the cubic term vanishes in the limit, the stated series is mathematically false and would teach a student incorrect coefficients.
gpt-oss:20b: fail (misleading) 2026-10-14 — Sentence 2 incorrectly sets up a limit with the wrong numerator and denominator; it is not needed for the correct result and could mislead a student.

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.