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Limits using Taylor series

Problem 7.676 · medium

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

Near 0, -x + sin(x) = x**5/120 - x**3/6 + ⋯ (its Maclaurin series).
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{5}}{120} + \frac{x^{3}}{6} - x + \sin{\left(x \right)}}{x^{5}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{- \frac{x^{5}}{60} + \frac{x^{3}}{3}}{x^{3}} = \frac{1}{3} - \frac{x^{2}}{60} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{2 x - 2 \sin{\left(x \right)}}{x^{3}}\right) = \frac{1}{3} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( \frac{1}{3} \)

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) — The solution incorrectly states the Maclaurin series for -x + sin(x) as starting with x^5/120, omitting the crucial -x^3/6 term. This error is masked by the fact that the final answer is correct, but the intermediate step (Line 1) is mathematically false and would teach a student the wrong expansion.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly states the Maclaurin series for -x + sin(x) as starting with x^5/120, omitting the crucial -x^3/6 term. This error is masked by the fact that the final answer is correct, but the intermediate step (Line 1) is mathematically false and would teach a student the wrong expansion.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the Maclaurin series for sin(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly expands the numerator as proportional to (-x + sin(x)), whereas the problem requires expanding (2x - 2sin(x)). This sign error leads to the wrong coefficient for the x^3 term, resulting in an incorrect limit of 1/3 instead of the correct value 1/3 (wait, 2*(1/6) = 1/3, so the magnitude is correct, but the sign in step 1 is wrong: -x + sin(x) has leading term -x^3/6, so 2*(-(-x^3/6)) = x^3/3? No. sin(x) ~ x - x^3/6. 2x - 2sin(x) ~ 2x - 2(x - x^3/6) = x^3/3. Limit is 1/3. Step 1 says -x + sin(x) = -x^3/6... Step 3 uses -2*(... - x^3/6) = 2x^3/6 = x^3/3. The algebra in step 3 accidentally corrects the sign error from step 1? Step 1: -x + sin(x) ~ -x^3/6. Step 3: -2 * (-x^3/6) = x^3/3. Denominator x^3. Result 1/3. The final answer is actually correct. However, Step 1 states '-x + sin(x) = x**5/120 - x**3/6'. The expansion of sin(x) is x - x^3/6 + x^5/120. So -x + sin(x) = -x^3/6 + x^5/120. Step 1 is algebraically correct. Step 3: Numerator is 2x - 2sin(x) = -2(-x + sin(x)). So -2 * (-x^3/6 + ...) = x^3/3 - ... Divide by x^3 gives 1/3. The logic holds. Why did I think it was wrong? Let's re-read carefully. Step 1: '-x + sin(x) = ...'. Step 3: '(-2*(x**5/120 - x**3/6))/(x**3)'. Note the order in Step 3's numerator: x**5/120 - x**3/6. This matches Step 1. -2 * (-x^3/6) = +x^3/3. Limit is 1/3. The solution is actually correct. Wait, is there a subtle error? Step 1 says 'Near 0, -x + sin(x) = ...'. This is a valid identity. Step 3 substitutes this into the expression for 2x - 2sin(x) by factoring out -2. 2x - 2sin(x) = -2(-x + sin(x)). This is correct. The limit calculation is correct. The answer 1/3 is correct. Is there a style issue? Step 2 is confusing but marked unchecked equation. Step 1 is a sentence. It is correct. Step 3 is an equation. Step 4 is an equation. The sentences are minimal. Step 1 is the only sentence. It is correct. Verdict: pass.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.