<p>If \(\lim_{x \to 0} \frac{x(1 + a\cos x) + b\sin x}{x^3} = 1\) then</p>
Step-by-Step Solution
Key Concept: For the limit to exist and equal 1 as x→0, the numerator must have a zero of order exactly 3 at x=0. Use Taylor expansion to match coefficients of powers of x systematically.
<p><strong>Step 1:</strong> Expand the numerator using Taylor series around x=0.</p><p>Numerator: x(1 + a cos x) + b sin x = x(1 + a(1 - x²/2 + x⁴/24 + ...)) + b(x - x³/6 + ...)</p><p>= x + ax - ax³/2 + bx - bx³/6 + ...</p><p>= (1 + a + b)x - (a/2 + b/6)x³ + ...</p><p><strong>Step 2:</strong> For the limit to exist and be finite, the coefficient of x must vanish (otherwise limit → ∞).</p><p>Coefficient of x: 1 + a + b = 0 ⟹ a + b = -1</p><p><strong>Step 3:</strong> With first-order term eliminated, the limit becomes:</p><p>lim(x→0) [-(a/2 + b/6)x³ + ...]/x³ = -(a/2 + b/6) = 1</p><p>So: a/2 + b/6 = -1</p><p><strong>Step 4:</strong> Solve the system:</p><p>• a + b = -1</p><p>• a/2 + b/6 = -1</p><p>From equation 1: b = -1 - a</p><p>Substitute: a/2 + (-1-a)/6 = -1</p><p>3a/6 - (1+a)/6 = -1</p><p>(3a - 1 - a)/6 = -1</p><p>(2a - 1)/6 = -1</p><p>2a - 1 = -6 ⟹ a = -5/2</p><p>Then: b = -1 - (-5/2) = 3/2</p><p>∴ Answer: C (a = -5/2, b = 3/2)</p>
Correct Answer: C