Home›Calculus 2›Limits using Taylor series›Problem 7.613

Limits using Taylor series

Problem 7.613 · medium

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

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
\[ \frac{- \frac{x^{4}}{12} + x^{2}}{x^{2}} = 1 - \frac{x^{2}}{12} \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{2 - 2 \cos{\left(x \right)}}{x^{2}}\right) = 1 \]
Let x → 0: only the constant term survives.✓ Proved

Answer \( 1 \)

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 2026-10-17 — 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 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the Maclaurin series for cosine, substitutes it into the limit expression, and simplifies to find the limit. The logic is sound and the steps are valid.
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 for cos(x) - 1; it states the x^2 term is -x^2/2, but the correct expansion is 1 - x^2/2 + ..., so cos(x) - 1 = -x^2/2 + ... The subsequent algebraic manipulation in Line 3 attempts to correct this by using the negative of the expression in Line 1, but the initial statement is factually incorrect and confusing.
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.