Limits of functions of two variables
Problem 10.640 · medium
Find \( \displaystyle \lim_{(x,y)\to(-2,1)} \frac{x^{2} + 3 x y + 2 y^{2}}{x + 2 y} \), 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
- \[ \left(x + y\right) \left(x + 2 y\right) = x^{2} + 3 x y + 2 y^{2} \]The numerator factored.✓ Proved
- \[ \left. x + y \right|_{\substack{ x=-2\\ y=1 }} = -1 \]Then substitute.✓ Proved
Answer \( -1 \)
✓ 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: passqwen3.6:27b-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: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly identifies the indeterminate form, factors the numerator, and cancels the common term validly for the limit process. The final substitution is correct.gpt-oss:20b: pass 2026-10-16qwen3.6:27b-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 2026-10-18 with SymPy 1.14.0.