Home›Calculus 3›Limits of functions of two variables›Problem 10.683

Limits of functions of two variables

Problem 10.683 · easy

Find \( \displaystyle \lim_{(x,y)\to(0,0)} - y + \cos{\left(x y \right)} \), or show it does not exist.

The function is continuous at the point (no division by zero there), so substitute.
Reviewed
\[ \left. - y + \cos{\left(x y \right)} \right|_{\substack{ x=0\\ y=0 }} = 1 \]
Direct substitution.✓ Proved

Answer \( 1 \)

✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	values along ten paths into the point (lines and parabolas) all approach it

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies that the function is continuous at the origin because it is composed of continuous functions (polynomials and cosine). Direct substitution is the valid method for continuous functions, and the calculation is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies that the function is continuous at the origin because it is composed of continuous functions (polynomials and cosine). Direct substitution is the valid method for continuous functions, and the calculation is correct.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies that the function is continuous at (0,0) and applies direct substitution, which is valid for continuous functions.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies that the function is continuous at (0,0) because it is composed of continuous functions (polynomials and cosine) and thus the limit can be found by direct substitution.
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/multivariable_limits, checked 2026-10-18 with SymPy 1.14.0.