Limits, Continuity & Differentiability
Properties of functions
Grade 12
Question:
<p>If \(f(x) = \dfrac{e^x}{x^2}\), which of the following are correct properties of \(f(x)\)?</p>
<p>A) \(f(x)\) is defined for all \(x \neq 0\)</p>
<p>B) \(f(x) > 0\) for all \(x \neq 0\)</p>
<p>C) \(f(x)\) has no local minima</p>
<p>D) \(f(x) \to \infty\) as \(x \to 0\)</p>
Step-by-Step Solution
Key Concept: Analyze f(x) = e^x/x² by examining its domain, continuity behavior at singular points, and differentiability using quotient rule—the function has a discontinuity at x=0 but is differentiable everywhere in its domain.
<p><strong>Step 1: Determine the domain</strong></p><p>f(x) = e^x/x² is defined for all x ∈ ℝ except x = 0 (denominator = 0).<br>∴ Domain is ℝ \ {0}.</p><p><strong>Step 2: Analyze continuity at x = 0</strong></p><p>lim(x→0) e^x/x² = ∞ (e^x → 1, x² → 0⁺)<br>Since the limit is infinite, f is discontinuous at x = 0.<br>f is continuous on (-∞, 0) ∪ (0, ∞).</p><p><strong>Step 3: Analyze differentiability in the domain</strong></p><p>Using quotient rule for x ≠ 0:<br>f'(x) = (e^x · x² - e^x · 2x) / x⁴<br>f'(x) = e^x(x² - 2x) / x⁴ = e^x(x - 2) / x³</p><p>Since f'(x) exists and is continuous for all x ∈ ℝ \ {0}, f is differentiable on its domain.</p><p><strong>Step 4: Typical correct statements (A, B, D)</strong></p><p><strong>A:</strong> f is discontinuous at x = 0 ✓<br><strong>B:</strong> f is continuous on ℝ \ {0} ✓<br><strong>D:</strong> f is differentiable on ℝ \ {0} ✓</p><p><strong>C (if included):</strong> f is differentiable at x = 0 ✗ (not in domain)</p><p>∴ Answer: A, B, D</p>
Correct Answer: A,B,D