<p>\(\lim_{x \to 0} \frac{x^2}{y} = 0\) when \((x, y) \to (0,0)\) along the curve \(y^2 = x^2\). State whether this is true or false.</p>
Step-by-Step Solution
Key Concept: The curve y² = x² represents y = x or y = -x (two lines through origin), not a single path. When substituting either form into the limit expression, the limit depends on which branch is taken, so the limit doesn't exist as a unique value.
<p><strong>Step 1:</strong> Analyze the curve y² = x².</p><p>This equation gives y = x or y = -x (two straight lines through the origin).</p><p><strong>Step 2:</strong> Check the limit along y = x.</p><p>Substituting y = x: lim(x→0) x²/x = lim(x→0) x = 0 ✓</p><p><strong>Step 3:</strong> Check the limit along y = -x.</p><p>Substituting y = -x: lim(x→0) x²/(-x) = lim(x→0) (-x) = 0 ✓</p><p><strong>Step 4:</strong> Verify with arbitrary path on y² = x².</p><p>For any point on y² = x², we have |y| = |x|, so |x²/y| = |x²/y| = |x|·|x/y| ≤ |x|·1 → 0.</p><p><strong>Conclusion:</strong> The limit equals 0 along all paths on the curve y² = x².</p><p>∴ The statement is <strong>TRUE</strong> (Answer: A)</p>
Correct Answer: A