Home›Calculus 2›Limits using Taylor series›Problem 7.604

Limits using Taylor series

Problem 7.604 · medium

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

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
\[ \frac{\frac{x^{4}}{12} - x^{2}}{x^{2}} = \frac{x^{2}}{12} - 1 \]
Divide through by the denominator.✓ Proved
\[ \lim_{x \to 0}\left(\frac{2 \cos{\left(x \right)} - 2}{x^{2}}\right) = -1 \]
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), substitutes it into the limit expression, simplifies the algebra, and evaluates the limit correctly. The intermediate step involving the error term (line 2) is technically valid as a justification for truncating the series, though slightly verbose.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the Maclaurin series for cos(x), substitutes it into the limit expression, simplifies the algebra, and evaluates the limit correctly. The intermediate step involving the error term (line 2) is technically valid as a justification for truncating the series, though slightly verbose.
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 intermediate step (line 2) is slightly unusual in its phrasing but mathematically valid as a confirmation of the series expansion order, and the final result is correct.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: inconclusive 2026-10-14 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "Line 1 contains a sign error in the Maclaurin series for cos(x) - 1; it should be -x^2/2 + x^4/24, not x^4/24 - x^2/2 (though the terms are li
gpt-oss:20b: inconclusive 2026-10-14 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The solution misapplies the Taylor expansion: it cancels terms incorrectly and uses a denominator of \(x^4\) instead of \(x^2\). The algebraic steps do no

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.