Home›Calculus 3›Limits of functions of two variables›Problem 10.587

Limits of functions of two variables

Problem 10.587 · medium

Find \( \displaystyle \lim_{(x,y)\to(-3,1)} \frac{x^{2} + 4 x y + 3 y^{2}}{x + 3 y} \), 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).
Reviewed
\[ \left(x + y\right) \left(x + 3 y\right) = x^{2} + 4 x y + 3 y^{2} \]
The numerator factored.✓ Proved
\[ \left. x + y \right|_{\substack{ x=-3\\ y=1 }} = -2 \]
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 indeterminate form, factors the numerator, cancels the common term (justified by the limit definition excluding the point itself), and evaluates the remaining continuous function.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the indeterminate form, factors the numerator, cancels the common term (justified by the limit definition excluding the point itself), and evaluates the remaining continuous function.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies the indeterminate form, factors the numerator to cancel the common term, and evaluates the limit of the simplified expression.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the indeterminate form, factors the numerator, cancels the common term (valid since the denominator is non-zero in a punctured neighborhood of the limit point), and evaluates the simplified expression.
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.