Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f(x)=x^2\sin(1/x)\)</span> (x≠0), f(0)=0. At x=0:</p>
f not cont
<strong>f cont, f' not cont</strong>
f and f' both cont
f' cont, f not
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>f continuous (squeeze). f'(0)=lim x sin(1/x)=0. f'(x)=2x sin(1/x)-cos(1/x): oscillates as x→0, not continuous. <strong>Answer: (2) f continuous and f' not continuous</strong></p><div class="key-concept"><strong>Key Concept:</strong> x²sin(1/x): differentiable at 0 but derivative discontinuous</div></div>
Correct Answer: 2