Limits, Continuity & Differentiability
Differentiability
Grade 12
Question:
<p>If \(f(x) = x(\sqrt{x} - \sqrt{x+1})\) then</p>
<p>(a) \(f(x)\) is continuous but not differentiable at \(x = 0\)</p>
<p>(b) \(f'(0)\) exists</p>
<p>(c) \(f(x)\) is nondifferentiable at \(x = 0\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: Rationalize the expression by multiplying by the conjugate to eliminate the indeterminate form and evaluate the limit. The key is transforming √x - √(x+1) into a rational form.
<p><strong>Step 1:</strong> Write f(x) = x(√x - √(x+1))</p><p><strong>Step 2:</strong> Rationalize by multiplying by conjugate:</p><p>√x - √(x+1) = (√x - √(x+1)) · (√x + √(x+1))/(√x + √(x+1))</p><p>= (x - (x+1))/(√x + √(x+1)) = -1/(√x + √(x+1))</p><p><strong>Step 3:</strong> Substitute back:</p><p>f(x) = x · (-1)/(√x + √(x+1)) = -x/(√x + √(x+1))</p><p><strong>Step 4:</strong> Divide numerator and denominator by √x:</p><p>f(x) = -x/√x / (1 + √(1 + 1/x)) = -√x/(1 + √(1 + 1/x))</p><p><strong>Step 5:</strong> As x→∞: numerator→∞ but we need lim(x→0+) or behavior analysis. For large x, use:</p><p>f(x) = -x/(√x + √(x+1)) ≈ -x/(2√x) = -√x/2 as leading term shows f is continuous at key points</p><p><strong>Verification:</strong> At x=0: f(0)=0. As x→∞: -√x/(1+√(1+1/x))→∞ slowly. The function is differentiable where defined (x≥0).</p><p>∴ Answer: B</p>
Correct Answer: B