Limits, Continuity & Differentiability
Continuity and Differentiability
Grade 12
Question:
<p>Consider the function \(f(x) = \begin{cases} \max\left\{x, \dfrac{1}{x}\right\}, & \text{when } x \neq 0 \\ \max\left\{x, \dfrac{1}{x}\right\}, & \\ 1, & \text{when } x = 0 \end{cases}\), then:</p>
<p>\(\lim_{x \to 0^+} f(x) \neq 0\)</p>
<p>\(\lim_{x \to 0^-} f(x) = 0\)</p>
<p>\(f(x)\) is continuous for all \(x \neq 0\)</p>
<p>\(f(x)\) is derivable for all \(x \neq 0\)</p>
Step-by-Step Solution
Key Concept: Analyze the max function by identifying where x and 1/x intersect (at x = ±1), then determine continuity and differentiability at critical points x = -1, 0, 1 by checking left/right limits and derivatives.
<p><strong>Step 1: Find where max{x, 1/x} changes definition.</strong></p><p>For x > 0: x = 1/x when x = 1. For x < 1 > 0: 1/x > x; for 0 < x < 1: 1/x > x; for x > 1: x > 1/x.</p><p>For x < 0: x = 1/x when x = -1. For x < -1: x < 1/x; for -1 < x < 0: x > 1/x.</p><p><strong>Step 2: Determine the explicit piecewise form.</strong></p><p>f(x) = {1/x for x ∈ (-1,0)∪(0,1); x for x ∈ (-∞,-1]∪[1,∞); 1 at x=0}</p><p><strong>Step 3: Check continuity at x = 0.</strong></p><p>lim(x→0+) max{x, 1/x} = lim(x→0+) 1/x = +∞ ≠ f(0) = 1. Function is <strong>discontinuous at x = 0</strong>.</p><p><strong>Step 4: Check continuity at x = ±1.</strong></p><p>At x = 1: lim(x→1-) 1/x = 1 = f(1) ✓ and lim(x→1+) x = 1 ✓ (continuous)</p><p>At x = -1: lim(x→-1-) x = -1 = f(-1) ✓ and lim(x→-1+) 1/x = -1 ✓ (continuous)</p><p><strong>Step 5: Check differentiability at x = ±1.</strong></p><p>At x = 1: Left derivative = 0 (from 1/x), Right derivative = 1 (from x). <strong>Not differentiable</strong>.</p><p>At x = -1: Left derivative = 1 (from x), Right derivative = -1 (from 1/x). <strong>Not differentiable</strong>.</p><p><strong>Step 6: Check differentiability elsewhere.</strong></p><p>Where f(x) = x or f(x) = 1/x individually, f is differentiable (on respective intervals not including corners).</p><p>∴ <strong>A:</strong> f is discontinuous at x = 0 (TRUE) | <strong>C:</strong> f is not differentiable at x = ±1 (TRUE)</p>
Correct Answer: A,C