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Limits using Taylor series

Problem 7.678 · medium

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

Near 0, cos(x) - 1 = x**4/24 - x**2/2 + ⋯ (its Maclaurin series).
Reviewed
\[ \lim_{x \to 0}\left(\frac{- \frac{x^{4}}{24} + \frac{x^{2}}{2} + \cos{\left(x \right)} - 1}{x^{4}}\right) = 0 \]
The polynomial matches the numerator to that order.✓ Proved
\[ \frac{\frac{x^{4}}{8} - \frac{3 x^{2}}{2}}{x^{2}} = \frac{x^{2}}{8} - \frac{3}{2} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{3 \cos{\left(x \right)} - 3}{x^{2}}\right) = - \frac{3}{2} \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( - \frac{3}{2} \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the Maclaurin series for cos(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the Maclaurin series for cos(x) and applies it to evaluate the limit. The algebraic manipulation and limit evaluation are correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: fail (error) 2026-10-16 — Line 1 contains a sign error in the Maclaurin series for cos(x) - 1; it states x**4/24 - x**2/2, but the correct expansion is -x**2/2 + x**4/24. Although Line 3 uses the correct terms, Line 1 is factually incorrect and misleading.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — Line 1 contains a sign error in the Maclaurin series for cos(x) - 1; the x^2 term should be -x^2/2, but the line lists it as -x^2/2 while listing the x^4 term as positive x^4/24 (which is correct for cos(x)-1? No, cos(x) = 1 - x^2/2 + x^4/24, so cos(x)-1 = -x^2/2 + x^4/24. The line says 'x**4/24 - x**2/2', which is actually correct. Wait, let me re-read carefully. Line 1: 'cos(x) - 1 = x**4/24 - x**2/2'. This is correct. Line 2 checks the remainder. Line 3 substitutes this into the expression. 3*(x^4/24 - x^2/2)/x^2 = 3*(x^2/24 - 1/2) = x^2/8 - 3/2. This is correct. Line 4 takes the limit. The logic is sound. Why did I think it was wrong? Let me check the sign of the x^4 term in cos(x). cos(x) = 1 - x^2/2! + x^4/4! - ... = 1 - x^2/2 + x^4/24. So cos(x) - 1 = -x^2/2 + x^4/24. The solution writes 'x**4/24 - x**2/2'. This is mathematically equivalent. The subsequent steps follow correctly. The verdict should be 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.