Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(f(x)=\begin{cases}x^2\cos(1/x) & x<0\\ 0 & x=0\\ x^2\sin(1/x) & x>0\end{cases}\)</span>. Which are correct?</p>
f cont not diff
<strong>f cont and diff at 0</strong>
f' cont not diff
<strong>f' discontinuous at 0</strong>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p><strong>f at x=0:</strong> lim |f(x)|≤x²→0. Continuous ✓. </p><p><strong>f'(0):</strong> lim h²cos(1/h)/h=lim h·cos(1/h)=0 (LHD); lim h²sin(1/h)/h=0 (RHD). f'(0)=0. <strong>Differentiable</strong> (B) ✓.</p><p>(A) f continuous not diff: FALSE — it is differentiable.</p><p><strong>f'(x) for x≠0:</strong><br>x<0: f'(x)=2x·cos(1/x)+sin(1/x)<br>x>0: f'(x)=2x·sin(1/x)-cos(1/x)<br>As x→0: the terms with sin(1/x) or cos(1/x) oscillate — f'(x) is NOT continuous at x=0. (C) FALSE, (D) f' is discontinuous at x=0 ✓.</p><p><strong>Answer: (B),(D)</strong></p><div class="key-concept"><strong>Key Concept:</strong> x²sin(1/x) is differentiable at 0 but its derivative is not continuous there</div></div>
Correct Answer: B,D