Limits, Continuity & Differentiability
Limits of functions involving inverse trigonometry and fractional part
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>?</p><p>[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 domain of f(x) requires simultaneous validity of inverse trigonometric functions (|1-{x}| ≤ 1), the denominator's square root ({x} > 0), and the denominator itself ({x} ≠ 1). This restricts {x} ∈ (0,1), making f discontinuous at integer points where {x} = 0.
<p><strong>Step 1: Determine the domain constraints</strong></p><p>For f(x) to be defined:</p><ul><li>cos⁻¹(1-{x}) requires: |1-{x}| ≤ 1 ⟹ 0 ≤ {x} ≤ 2 (always satisfied since {x} ∈ [0,1))</li><li>sin⁻¹(1-{x}) requires: |1-{x}| ≤ 1 ⟹ 0 ≤ {x} ≤ 2 (always satisfied)</li><li>√(2{x}) requires: {x} > 0</li><li>(1-{x}) ≠ 0 requires: {x} ≠ 1 (automatically satisfied since {x} ∈ [0,1))</li></ul><p><strong>Step 2: Identify domain of f</strong></p><p>Domain: {x ∈ ℝ : {x} ∈ (0,1)} = ℝ \ ℤ (all real numbers except integers)</p><p><strong>Step 3: Analyze continuity at integer points</strong></p><p>At x = n (integer): lim(x→n⁻) f(x) exists as {x}→1⁻, but lim(x→n⁺) f(x) has {x}→0⁺, and f(n) is undefined. Therefore, f is discontinuous at all integers.</p><p><strong>Step 4: Verify differentiability</strong></p><p>Since f is discontinuous at every integer point, f is not differentiable at any integer x ∈ ℤ. Within each interval (n, n+1), f is continuous and differentiable.</p><p><strong>Answer: A</strong> (f is discontinuous at integer points) <strong>and C</strong> (f is not differentiable at integer points)</p>
Correct Answer: AC