Home›Calculus 3›Limits of functions of two variables›Problem 10.781

Limits of functions of two variables

Problem 10.781 · easy

Find \( \displaystyle \lim_{(x,y)\to(3,3)} \frac{x^{2} + y}{x y + 1} \), or show it does not exist.

The function is continuous at the point (no division by zero there), so substitute.
Reviewed
\[ \left. \frac{x^{2} + y}{x y + 1} \right|_{\substack{ x=3\\ y=3 }} = \frac{6}{5} \]
Direct substitution.✓ Proved

Answer \( \frac{6}{5} \)

✓ 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 point (3,3) because the denominator is non-zero, and correctly applies direct substitution.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-11-02 — The solution correctly identifies that the function is continuous at the point (3,3) because the denominator is non-zero, and correctly applies direct substitution.
gpt-oss:20b: pass 2026-11-02
qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly identifies that the function is continuous at the point (3,3) because the denominator is non-zero, justifying direct substitution.
gpt-oss:20b: pass 2026-11-01
qwen3.6:27b-mlx: pass 2026-11-01 — The solution correctly identifies that the function is continuous at the point (3,3) because the denominator is non-zero, and correctly applies direct substitution to find the limit.
gpt-oss:20b: pass 2026-11-01

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 2026-11-02 with SymPy 1.14.0.