Home›Calculus 2›Limits using Taylor series›Problem 7.708

Limits using Taylor series

Problem 7.708 · 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
\[ \frac{- \frac{x^{3}}{3} + \frac{x^{2}}{2}}{x^{2}} = \frac{1}{2} - \frac{x}{3} \]
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: it states -x + log(x+1) equals a series starting with x^3/3, but the correct expansion is -x^2/2 + x^3/3 - ... (the x^2 term is missing and the sign of the leading term is wrong relative to the LHS). This fundamental algebraic error invalidates the subsequent limit calculation, although the final answer happens to be correct by coincidence of the error cancellation in step 3.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-19 — Step 1 contains a sign error in the Maclaurin series expansion: it states -x + log(x+1) equals a series starting with x^3/3, but the correct expansion is -x^2/2 + x^3/3 - ... (the x^2 term is missing and the sign of the leading term is wrong relative to the LHS). This fundamental algebraic error invalidates the subsequent limit calculation, although the final answer happens to be correct by coincidence of the error cancellation in step 3.
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-16 — Step 1 contains a sign error in the Maclaurin series expansion: it claims -x + log(x+1) = x^3/3 - x^2/2 + ..., but the correct expansion is -x^2/2 + x^3/3 - ... (the x^2 term is negative, not positive as implied by the order of terms if interpreted strictly, but more importantly, the subsequent steps rely on the numerator being x - log(x+1), which is x^2/2 - x^3/3 + ...). Step 3 divides -(x^3/3 - x^2/2) by x^2 to get 1/2 - x/3, which is algebraically correct for that expression, but Step 1's assertion is confusingly written and potentially misleading regarding the sign of the leading term relative to the problem statement. Specifically, the problem asks for (x - ln(x+1))/x^2. The series for x - ln(x+1) is x^2/2 - x^3/3 + ... . Step 1 writes -x + log(x+1) which is the negative of the numerator. The series for -x + log(x+1) is -x^2/2 + x^3/3 - ... . Step 1 says it equals x^3/3 - x^2/2 + ... which is actually correct (-x^2/2 + x^3/3). However, Step 3 uses -(x^3/3 - x^2/2) which is -x^3/3 + x^2/2. This is the series for x - log(x+1). So Step 3 effectively switches back to the positive numerator. The logic is disjointed: Step 1 expands the negative of the numerator, Step 3 negates the result of Step 1 (implicitly) to get the positive numerator's expansion, then divides. While the final answer is correct, the presentation in Step 1 and the transition to Step 3 is confusing and contains a sign flip that is not explicitly justified, making it a 'misleading' or 'error' in pedagogical clarity. Let's look closer. Step 1: -x + log(x+1) = -x^2/2 + x^3/3 ... Step 1 writes: x^3/3 - x^2/2. This is correct. Step 3: -(x^3/3 - x^2/2)/x^2. This is -(-x^2/2 + x^3/3)/x^2 = (x^2/2 - x^3/3)/x^2. This matches the numerator of the problem (x - log(x+1)). So the math works out. Is it misleading? It's very confusing to expand the negative of the numerator and then negate it in the next step without explanation. A student might think the series for x - log(x+1) is x^3/3 - x^2/2. That would be wrong. The series for x - log(x+1) is x^2/2 - x^3/3. Step 1 says -x + log(x+1) is x^3/3 - x^2/2. This is correct. But a student reading quickly might associate the RHS with the problem's numerator. This is misleading.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution incorrectly expands the numerator as -x + log(x+1) instead of x - log(x+1), leading to the wrong sign for the x^2 term. Consequently, it calculates the limit as 1/2 instead of the correct value -1/2.
gpt-oss:20b: pass 2026-10-16

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 2026-10-19 with SymPy 1.14.0.