Limits, Continuity & Differentiability
General
Grade 12
Question:
<p><span class="math-inline">\(\phi(x)=[|x|-|\sin x|]\)</span> (where [.] is GIF). Which are correct?</p>
<strong>derivable at x=0</strong>
<strong>continuous at x=0</strong>
lim φ(x) DNE
<strong>cont and derivable at x=0</strong>
Step-by-Step Solution
Key Concept: General
<div class="solution"><p>Near x=0: <span class="math-inline">$|x|-|\sin x|\ge 0$</span> and <span class="math-inline">$\to 0$</span>. For small x, <span class="math-inline">$|x|>|\sin x|$</span>, so <span class="math-inline">$|x|-|\sin x|\in(0,1)$</span> for small x≠0. Thus <span class="math-inline">$[|x|-|\sin x|]=0$</span> near 0, same as φ(0)=0. Continuous at x=0 ✓(B).</p><p>(A) Derivable at x=0: φ=0 near 0, so φ'(0)=0. Derivable ✓(A).</p><p>(C) lim φ(x) does not exist: limit IS 0. FALSE.</p><p>(D) Continuous and derivable at x=0: Both TRUE ✓(D).</p><p><strong>Answer: (A),(B),(D)</strong></p><div class="key-concept"><strong>Key Concept:</strong> |x|>|sin x| for small x≠0 (since sin x < x for x>0)</div></div>
Correct Answer: A,B,D