Question:

<p><strong>Step 1:</strong> Analyze from the right (x → 0⁺): |x| = x, so</p><p>lim(x→0⁺) (1-e^x)·sin(x)/x = (1-1)·1 = 0, but we need the rate.</p><p><strong>Step 2:</strong> Use Taylor expansions: e^x = 1 + x + x²/2 + ..., sin(x) = x - x³/6 + ...</p><p>So (1-e^x)·sin(x)/x = -(x + x²/2 + ...)·(1 - x²/6 + ...)</p><p>As x → 0⁺: (1-e^x)·sin(x)/x → -x (approaching 0 from below, ≈ -0.something)</p><p><strong>Step 3:</strong> More precisely, (1-e^x)·sin(x)/|x| = -(1 - x/2 + ...)·(1 - x²/6 + ...) → -1 + O(x) as x → 0⁺</p><p>For small positive x, the expression ≈ -1 + small positive term, so values lie in (-1, 0).</p><p><strong>Step 4:</strong> From the left (x → 0⁻): |x| = -x, so</p><p>(1-e^x)·sin(x)/(-x) = (1-e^x)·(-sin(x)/x) → 1 + small term, staying in (0, 1).</p><p><strong>Step 5:</strong> Since the expression approaches values in (-1, 0) from the right and (0, 1) from the left, as x → 0 the function oscillates between values in (-1, 1) but never reaches exactly ±1.</p><p>Therefore ⌊(1-e^x)·sin(x)/|x|⌋ = <strong>-1</strong> for x → 0⁺ and <strong>0</strong> for x → 0⁻.</p><p>The limit does not exist in classical sense, but evaluating the principal behavior near 0: the answer is <strong>-1</strong>.</p><p>∴ Answer: B</p>