Limits, Continuity & Differentiability
Limits
Grade 12
Question:
<p>If \(f(x) = \frac{\sin[x]}{[x]}, [x] \neq 0\)<br>\(= 0, [x] = 0\)<br>where \([x]\) is the greatest integer function, then \(\lim_{x \to 0} f(x)\) equals:</p>
<p>(a) 1</p>
<p>(b) 0</p>
<p>(c) \(-1\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: As x → 0, [x] takes only the values 0 and -1 (never positive integers), so we must evaluate left and right limits separately. The right limit approaches 0 because [x] = 0 for 0 ≤ x < 1, while the left limit requires analyzing [x] = -1 for -1 ≤ x < 0.
<p><strong>Step 1:</strong> Analyze right limit as x → 0⁺</p><p>For 0 ≤ x < 1: [x] = 0, so f(x) = 0 (by definition when [x] = 0)</p><p>Therefore: lim(x → 0⁺) f(x) = 0</p><p><strong>Step 2:</strong> Analyze left limit as x → 0⁻</p><p>For -1 ≤ x < 0: [x] = -1, so f(x) = sin(-1)/(-1) = -sin(1)/(-1) = sin(1)</p><p>Therefore: lim(x → 0⁻) f(x) = sin(1)</p><p><strong>Step 3:</strong> Check if limit exists</p><p>Since lim(x → 0⁺) f(x) = 0 ≠ sin(1) = lim(x → 0⁻) f(x), the left and right limits are unequal.</p><p><strong>Step 4:</strong> Conclude</p><p>The limit does not exist because one-sided limits are different. The answer is <strong>D (Does not exist)</strong>.</p>
Correct Answer: D