Limits, Continuity & Differentiability
Limits involving greatest integer function
Grade 12
Question:
<p>The value of \(\displaystyle\lim_{x\to 0}\left\lfloor (1-e^x)\frac{\sin x}{|x|}\right\rfloor\) equals:<br><b>[Note:</b> \([\,\cdot\,]\) denotes the greatest integer function.<b>]</b></p>
<p>0</p>
<p>\(-1\)</p>
<p>1</p>
<p>does not exist</p>
Step-by-Step Solution
Key Concept: Analyze the limit of (1-e^x)·(sin x/|x|) separately from both sides since |x| creates a piecewise behavior, then apply the greatest integer function to the resulting limit value.
<p><strong>Step 1:</strong> Analyze the limit as x→0⁺ and x→0⁻ separately due to |x|.</p><p><strong>Step 2:</strong> For x→0⁺: |x| = x, so we have lim(x→0⁺) (1-e^x)·(sin x/x) = (1-1)·1 = 0.</p><p><strong>Step 3:</strong> For x→0⁻: |x| = -x, so we have lim(x→0⁻) (1-e^x)·(sin x/(-x)) = (1-1)·(-1) = 0.</p><p><strong>Step 4:</strong> Both sided limits equal 0, so lim(x→0) (1-e^x)·(sin x/|x|) = 0.</p><p><strong>Step 5:</strong> Apply the greatest integer function: ⌊0⌋ = 0.</p><p>∴ Answer: <strong>B (which is 0)</strong></p>
Correct Answer: B