Home›Calculus 3›Limits of functions of two variables›Problem 10.646

Limits of functions of two variables

Problem 10.646 · medium

Find \( \displaystyle \lim_{(x,y)\to(3,1)} \frac{x^{2} - 2 x y - 3 y^{2}}{x - 3 y} \), 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).
\[ \left(x - 3 y\right) \left(x + y\right) = x^{2} - 2 x y - 3 y^{2} \]
The numerator factored.✓ Proved
\[ \left. x + y \right|_{\substack{ x=3\\ y=1 }} = 4 \]
Then substitute.✓ Proved

Answer \( 4 \)

Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	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: fail (misleading) — The sentence claims the common factor is nonzero near the point, which is false because along the line x=3y (through (3,1)) the factor vanishes arbitrarily close to the point. While the cancellation still yields the correct limit, the justification given is misleading.
qwen3.6:27b-mlx: pass — The solution correctly identifies the removable discontinuity, factors the numerator, cancels the common term, and evaluates the limit of the simplified expression.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the removable discontinuity, factors the numerator, cancels the common term, and evaluates the limit of the simplified expression.
gpt-oss:20b: fail (misleading) 2026-10-18 — The sentence claims the common factor is nonzero near the point, which is false because along the line x=3y (through (3,1)) the factor vanishes arbitrarily close to the point. While the cancellation still yields the correct limit, the justification given is misleading.
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the removable discontinuity, factors the numerator, and cancels the common term to evaluate the limit.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, factors the numerator to cancel the common term, and evaluates the limit of the simplified expression. The reasoning is sound.
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.