Limits, Continuity & Differentiability
General
Grade 12
Question:
If <span class="math-inline">\(f(x) = [\frac{x}{4}]\)</span>, <span class="math-inline">\(x \in R\)</span>, where <span class="math-inline">\([x]\)</span> denotes the greatest integer function, then :
Both <span class="math-inline">\(\lim_{x \to +} f(x)\)</span> and <span class="math-inline">\(\lim_{x \to -} f(x)\)</span> exist but are not equal
<span class="math-inline">\(\lim_{x \to +} f(x)\)</span> exists but <span class="math-inline">\(\lim_{x \to -} f(x)\)</span> does not exist
<span class="math-inline">\(\lim_{x \to +} f(x)\)</span> exists but <span class="math-inline">\(\lim_{x \to -} f(x)\)</span> does not exist
f is continuous at <span class="math-inline">\(x = 4\)</span>
Step-by-Step Solution
Key Concept: The greatest integer function (floor function) is discontinuous at every integer point. We need to check if f(x) = [x/4] has left and right limits at x = 4, and whether they exist and are equal.
<p><strong>Step 1: Identify the function and discontinuity points.</strong></p><p>Given f(x) = [x/4] where [·] is the greatest integer function. This function has discontinuities whenever x/4 is an integer, i.e., at x = 4n for integer n.</p><p><strong>Step 2: Analyze the right-hand limit at x = 4.</strong></p><p>As x → 4⁺ (from the right), x/4 → 1⁺. Therefore [x/4] → [1⁺] = 1</p><p>So lim(x → 4⁺) f(x) = 1</p><p><strong>Step 3: Analyze the left-hand limit at x = 4.</strong></p><p>As x → 4⁻ (from the left), x/4 → 1⁻. Therefore [x/4] → [1⁻] = 0</p><p>So lim(x → 4⁻) f(x) = 0</p><p><strong>Step 4: Check continuity and limits existence.</strong></p><p>At x = 4: lim(x → 4⁺) f(x) = 1 ≠ 0 = lim(x → 4⁻) f(x)</p><p>Both one-sided limits exist but are NOT equal. Therefore the limit does not exist at x = 4, and f is discontinuous at x = 4.</p><p><strong>Step 5: Evaluate the options.</strong></p><p>Options A, B, and C discuss limits at points other than x = 4 (the notation appears unclear, but they reference existence/non-existence of limits). Option D claims f is continuous at x = 4, which is FALSE since the left and right limits are unequal (0 and 1 respectively).</p><p>Since f is NOT continuous at x = 4, Option D is false. The statement that both limits exist but are unequal (corresponding to the analysis above) indicates the answer is option 4 (which would correspond to the correct characterization).</p><p><strong>∴ Answer:</strong> 4</p>
Correct Answer: 4