Home›Calculus 3›Limits of functions of two variables›Problem 10.584

Limits of functions of two variables

Problem 10.584 · medium

Find \( \displaystyle \lim_{(x,y)\to(0,0)} \frac{x y}{x^{2} + y^{2}} \), or show it does not exist.

Approach (0, 0) along the curves y = m*x for different m.
Reviewed
\[ \lim_{x \to 0} 0 = 0 \]
Along y = 0.✓ Proved
\[ \lim_{x \to 0} \frac{1}{2} = \frac{1}{2} \]
Along y = x: a different value, so the limit does not exist.✓ Proved

Answer \( \text{does not exist} \)

✓ Nihil obstat Lines: 2 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
3	✓ 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) disagree

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies that the limit depends on the path of approach by comparing the limits along y=0 and y=x. Since the limits are different (0 vs 1/2), the conclusion that the limit does not exist is correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies that the limit depends on the path of approach by comparing the limits along y=0 and y=x. Since the limits are different (0 vs 1/2), the conclusion that the limit does not exist is correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly applies the path test for non-existence of limits. It identifies two paths (y=0 and y=x) yielding different limits (0 and 1/2), which is sufficient to prove the limit does not exist.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies that the limit depends on the path of approach (specifically comparing y=0 and y=x), which is sufficient to prove the limit does not exist.
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/multivariable_limits, checked 2026-10-17 with SymPy 1.14.0.