Limits, Continuity & Differentiability
Limits using functional equations
Grade 12
Question:
<p><strong>997.</strong> Let <em>f</em> be a real valued derivable function such that <em>f</em>(<em>x</em>)<em>f</em>(<em>y</em>) = <em>f</em>(<em>x</em>)<em>y</em> + <em>x</em><em>f</em>(<em>y</em>), ∀<em>x</em>, <em>y</em> ∈ ℝ. If <em>f</em>′(0) = 2, then find \(\lim_{x \to 0} \left[\dfrac{f(x)}{\sin x}\right]\). [Note: [ ] represents greatest integer function.]</p>
Step-by-Step Solution
Key Concept: Use the functional equation f(x)f(y) = f(x)y + xf(y) to find f(x) explicitly by differentiating or substituting strategic values, then evaluate the limit of f(x)/sin(x).
<p><strong>Step 1:</strong> Find f(x) from the functional equation.</p><p>Given: f(x)f(y) = f(x)y + xf(y)</p><p>Substitute y = 0: f(x)f(0) = f(x)·0 + x·f(0)</p><p>This gives: f(x)f(0) = xf(0)</p><p>If f(0) ≠ 0, then f(x) = x, but f'(x) = 1, contradicting f'(0) = 2.</p><p>So f(0) = 0.</p><p><strong>Step 2:</strong> Differentiate the functional equation with respect to y.</p><p>f(x)f'(y) = f(x) + xf'(y)</p><p>At y = 0: f(x)f'(0) = f(x) + xf'(0)</p><p>f(x)·2 = f(x) + x·2</p><p>2f(x) - f(x) = 2x</p><p>f(x) = 2x</p><p><strong>Step 3:</strong> Verify: f(x)f(y) = 2x·2y = 4xy and f(x)y + xf(y) = 2xy + 2xy = 4xy ✓</p><p><strong>Step 4:</strong> Evaluate the limit.</p><p>$$\lim_{x \to 0} \frac{f(x)}{\sin x} = \lim_{x \to 0} \frac{2x}{\sin x} = 2 \cdot \lim_{x \to 0} \frac{x}{\sin x} = 2 · 1 = 2$$</p><p><strong>Step 5:</strong> Apply greatest integer function.</p><p>$$\left[\lim_{x \to 0} \frac{f(x)}{\sin x}\right] = [2] = 2$$</p><p>∴ <strong>Answer: 2</strong></p>
Correct Answer: 2