Home›Calculus 2›Limits using Taylor series›Problem 7.611

Limits using Taylor series

Problem 7.611 · medium

Use Taylor series to evaluate \( \displaystyle \lim_{x\to 0} \frac{3 x - 3 \ln{\left(x + 1 \right)}}{x^{2}} \).

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
\[ \frac{- x^{3} + \frac{3 x^{2}}{2}}{x^{2}} = \frac{3}{2} - x \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{3 x - 3 \ln{\left(x + 1 \right)}}{x^{2}}\right) = \frac{3}{2} \]
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: pass
qwen3.6:27b-mlx: fail (error) — Line 1 states an incorrect Maclaurin series for -x + log(x+1); the x^2 term is -1/2, not -1/2 in the wrong sign context or missing terms, but specifically it claims the series starts with x^3/3, ignoring the x^2 term which is crucial for the limit. The series for log(1+x) is x - x^2/2 + x^3/3 - ..., so -x + log(1+x) = -x^2/2 + x^3/3 - ... . The solution incorrectly lists the leading term as x^3/3 (or implies the x^2 term is negligible/wrongly signed in the text representation). Actually, looking closely at line 1: "-x + log(x + 1) = x**3/3 - x**2/2 + ...". This is algebraically correct as a set of terms, but usually series are written in ascending order of powers. However, the bigger issue is Line 2. Line 2 computes a limit of a remainder term divided by x^3, which is irrelevant to the problem asking for division by x^2. The solution jumps from an incorrect/irrelevant intermediate step to the final answer. The logic is disjointed and the series expansion in line 1 is presented in a confusing order that suggests the x^2 term might be higher order or handled incorrectly in the student's mind, although the equation in line 3 uses the correct -x^2/2 term. The main error is that Line 2 is completely irrelevant to the problem statement (dividing by x^3 instead of x^2) and Line 1 is poorly formatted/misleading.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-17 — Line 1 states an incorrect Maclaurin series for -x + log(x+1); the x^2 term is -1/2, not -1/2 in the wrong sign context or missing terms, but specifically it claims the series starts with x^3/3, ignoring the x^2 term which is crucial for the limit. The series for log(1+x) is x - x^2/2 + x^3/3 - ..., so -x + log(1+x) = -x^2/2 + x^3/3 - ... . The solution incorrectly lists the leading term as x^3/3 (or implies the x^2 term is negligible/wrongly signed in the text representation). Actually, looking closely at line 1: "-x + log(x + 1) = x**3/3 - x**2/2 + ...". This is algebraically correct as a set of terms, but usually series are written in ascending order of powers. However, the bigger issue is Line 2. Line 2 computes a limit of a remainder term divided by x^3, which is irrelevant to the problem asking for division by x^2. The solution jumps from an incorrect/irrelevant intermediate step to the final answer. The logic is disjointed and the series expansion in line 1 is presented in a confusing order that suggests the x^2 term might be higher order or handled incorrectly in the student's mind, although the equation in line 3 uses the correct -x^2/2 term. The main error is that Line 2 is completely irrelevant to the problem statement (dividing by x^3 instead of x^2) and Line 1 is poorly formatted/misleading.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Step 1 states an incorrect Maclaurin series for -x + log(x+1), claiming the leading term is x^3/3 instead of -x^2/2. This fundamental error in the series expansion invalidates the subsequent limit calculation, even though the final numerical answer happens to be correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The Maclaurin series expansion in step 1 is incorrect; the expansion of log(x+1) is x - x^2/2 + x^3/3 - ..., so -x + log(x+1) equals -x^2/2 + x^3/3 - ..., not x^3/3 - x^2/2 + ... (the signs are swapped for the leading terms, although the final numerical result happens to be correct due to a compensating error in the subsequent algebraic manipulation, the stated series is factually wrong).
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.