Home›Calculus 3›Limits of functions of two variables›Problem 10.593

Limits of functions of two variables

Problem 10.593 · medium

Find \( \displaystyle \lim_{(x,y)\to(0,0)} \frac{x^{2} y}{x^{4} + y^{2}} \), or show it does not exist.

Approach (0, 0) along the curves y = m*x**2 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**2: 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 testing specific curves, and the algebraic results for those paths are 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 testing specific curves, and the algebraic results for those paths are correct.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies that the limit depends on the path of approach. By showing that the limit is 0 along y=0 and 1/2 along y=x^2, it validly proves 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. By showing different limits along y=0 and y=x^2, it rigorously proves 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.