Home›Calculus 3›Limits of functions of two variables›Problem 10.643

Limits of functions of two variables

Problem 10.643 · medium

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

Substitution gives 0/0. Factor the numerator and cancel the common factor (it is nonzero near, but not at, the point).
Reviewed
\[ \left(x - y\right) \left(x + y\right) = x^{2} - y^{2} \]
The numerator factored.✓ Proved
\[ \left. x + y \right|_{\substack{ x=1\\ y=1 }} = 2 \]
Then substitute.✓ Proved

Answer \( 2 \)

✓ 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) all approach it

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the removable discontinuity, factors the expression, and evaluates the limit of the simplified function. The logic is sound and complete for this problem.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the removable discontinuity, factors the expression, and evaluates the limit of the simplified function. The logic is sound and complete for this problem.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the removable discontinuity, factors the expression, and evaluates the limit of the simplified function.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the removable discontinuity, factors the expression, and evaluates the limit of the simplified function.
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 2026-10-18 with SymPy 1.14.0.