Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f(x)=x^2|\cos(\pi/x)|\)</span> (x≠0), f(0)=0. f is:</p>
diff at 0 and 2
diff at 0, not at 2
not at 0, diff at 2
neither
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>At x=0: f'(0)=lim x|cos(π/x)|=0 ✓. At x=2: |cos(π/2)|=0, corner from |cos|. Non-diff at x=2. <strong>Answer: (B) diff at 0, not diff at 2</strong></p><div class="trap-box"><strong>Trap:</strong> x² smooths oscillation at 0 but not at zeros of cos.</div><div class="key-concept"><strong>Key Concept:</strong> x²·bounded = differentiable at 0; |cos| has corner at its zeros</div></div>
Correct Answer: 2