Limits of functions of two variables
Problem 10.585 · medium
Find \( \displaystyle \lim_{(x,y)\to(0,0)} \frac{x^{2} - y^{2}}{x^{2} + y^{2}} \), or show it does not exist.
- Approach (0, 0) along the curves y = m*x for different m.
- \[ \lim_{x \to 0} 1 = 1 \]Along y = 0.✓ Proved
- \[ \lim_{x \to 0} 0 = 0 \]Along y = x: a different value, so the limit does not exist.✓ Proved
Answer \( \text{does not exist} \)
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) disagree |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the limit along y=x is 0, but substituting y=x into the expression yields (x^2 - x^2)/(x^2 + x^2) = 0, which is correct for that
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-10-17 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution claims the limit along y=x is 0, but substituting y=x into the expression yields (x^2 - x^2)/(x^2 + x^2) = 0, which is correct for thatgpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly identifies that the limit depends on the path of approach. By showing different limits along y=0 and y=x, it validly concludes the limit does not exist.gpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution claims the limit along y=x is 0, but substituting y=x into the expression yields (x^2 - x^2)/(x^2 + x^2) = 0, so the limit is indeed 0. However, the first line claims the limit along y=0 is 1. Substituting y=0 yields x^2/x^2 = 1, so that limit is 1. Since 1 != 0, the limit does not exist. The logic is actually correct, but the labeling in line 3 is confusing/wrongly phrased as 'Limit(0, x, 0, dir='+-') = 0 -- Along y = x'. Wait, let me re-read carefully. Line 2 says 'Limit(1, x, 0, dir='+-') = 1 -- Along y = 0'. This is correct. Line 3 says 'Limit(0, x, 0, dir='+-') = 0 -- Along y = x'. This is also correct. The conclusion 'a different value, so the limit does not exist' is correct. Why did I think it was wrong? Ah, I need to check if the *method* is sufficient. Approaching along lines y=mx is a standard way to show non-existence if the limit depends on m. Here, for y=mx, the expression is (1-m^2)/(1+m^2). For m=0, limit is 1. For m=1, limit is 0. Since the limits are different, the overall limit does not exist. The solution is actually correct. Let me double check the 'unchecked' status. The prompt says 'Each line of the solution is either an EQUATION, which a computer algebra system has already checked... or a SENTENCE'. Lines 2 and 3 are marked [equation, unchecked]. This is contradictory. If they are equations, they should be checked. If they are unchecked, they might be sentences or just raw data. But the prompt says 'do not re-check the algebra of a line marked proved'. These are marked 'unchecked'. This implies I should check them? Or does 'unchecked' mean the CAS didn't check them? The prompt says 'a computer algebra system has already checked (its result is shown; do not re-check the algebra of a line marked proved)'. It doesn't explicitly say what to do with 'unchecked' lines. However, usually 'unchecked' in these contexts implies the user provided the result and I should verify it or it's a sentence. Let's look at the content. Line 2: Limit is 1 along y=0. Correct. Line 3: Limit is 0 along y=x. Correct. Conclusion: DNE. Correct. Is there a subtle error? No. The solution is sound. Wait, is there a case where approaching along lines is not enough? No, if limits along different paths are different, the limit DNE. The solution is correct. Why would it fail? Maybe the notation 'Limit(1, x, 0, dir='+-')' is not standard? Or maybe the prompt implies that 'unchecked' lines are sentences? If they are sentences, they are factual claims. They are true. I will mark it as pass.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.