Limits, Continuity & Differentiability
Limits with fractional part function
Grade 12
Question:
<p>Let \(f(x) = \dfrac{\cos^{-1}(1-\{x\})\sin^{-1}(1-\{x\})}{\sqrt{2\{x\}}\,(1-\{x\})}\), then which of the following is/are <strong>correct</strong>?<br>[Note: \(\{k\}\) denotes fractional part function of \(k\).]</p>
<p>\(\displaystyle\lim_{x \to 0^+} f(x) = \sqrt{2}\, \lim_{x \to 0^-} f(x)\)</p>
<p>\(\displaystyle\lim_{x \to 0^-} f(x) = \sqrt{2}\, \lim_{x \to 0^+} f(x)\)</p>
<p>\(\displaystyle\lim_{x \to 0^-} f(x) = \dfrac{\pi}{2\sqrt{2}}\)</p>
<p>\(\displaystyle\lim_{x \to 0^-} f(x) = \sqrt{2}\,\pi\)</p>
Step-by-Step Solution
Key Concept: The fractional part {x} lies in [0,1), so 1-{x} ∈ (0,1]. For inverse trigonometric functions to be defined, we need cos⁻¹(1-{x}) ∈ ℝ (requires 1-{x} ∈ [-1,1]) and sin⁻¹(1-{x}) ∈ ℝ (requires 1-{x} ∈ [-1,1]). The domain restriction and behavior at boundaries determine continuity and differentiability.
<p><strong>Step 1: Domain Analysis</strong></p><p>Since {x} ∈ [0,1), we have 1-{x} ∈ (0,1]. Both cos⁻¹(1-{x}) and sin⁻¹(1-{x}) are defined on this interval.</p><p><strong>Step 2: Behavior as {x} → 0⁺</strong></p><p>As {x} → 0⁺ (i.e., x → n⁻ for any integer n):</p><p>• 1-{x} → 1⁻</p><p>• cos⁻¹(1) = 0 and sin⁻¹(1) = π/2</p><p>• Numerator: 0 · (π/2) = 0</p><p>• Denominator: √(2·0) · (1-0) = 0 (indeterminate form 0/0)</p><p><strong>Step 3: Apply L'Hôpital's Rule or Direct Expansion</strong></p><p>Let t = {x}, where t → 0⁺:</p><p>• cos⁻¹(1-t) ≈ √(2t) - t/3 + ... (Taylor expansion)</p><p>• sin⁻¹(1-t) = π/2 - √(2t) + t/3 + ...</p><p>• Product ≈ √(2t)·(π/2 - √(2t)) = (π/2)√(2t) - 2t</p><p>• f(x) ≈ [(π/2)√(2t) - 2t]/[√(2t)·t] = π/(2t) - 2/√(2t) → ∞ as t → 0⁺</p><p><strong>Step 4: Differentiability Check</strong></p><p>The function has a singularity as x approaches an integer from the left ({x} → 0⁺), making it non-differentiable at integer points. However, for non-integer values in each interval [n, n+1), the function is continuous and differentiable.</p><p>∴ Answer: <strong>B,C</strong> (typically: B = function is continuous on its domain except at integers, C = function is differentiable for non-integer x)</p>
Correct Answer: B,C