Limits, Continuity & Differentiability
Continuity and the Greatest Integer Function
Grade 12
Question:
<p><strong>Example 45:</strong> Let \([\cdot]\) denote the greatest integer function and \(f(x) = [\tan^2 x]\). Which of the following is true? [IIT JEE 1993]</p>
<p>(a) \(\lim_{x \to 0} f(x)\) doesn't exist</p>
<p>(b) \(f(x)\) is continuous at \(x = 0\)</p>
<p>(c) \(f(x)\) is not differentiable at \(x = 0\)</p>
<p>(d) \(f'(0) = 2\)</p>
Step-by-Step Solution
Key Concept: The greatest integer function applied to small values near zero yields zero, making the function continuous at that point.
<p><strong>Step 1:</strong> Consider $f(x) = [\tan^2 x]$ where $[\cdot]$ is the greatest integer function.</p><p><strong>Step 2:</strong> At $x = 0$: $\tan^2(0) = 0$, so $f(0) = [0] = 0$.</p><p><strong>Step 3:</strong> For small values of $x$ near 0, $\tan^2 x \approx x^2$ which is very small, so $[\tan^2 x] = 0$.</p><p><strong>Step 4:</strong> Thus $\lim_{x \to 0} f(x) = 0 = f(0)$, so $f(x)$ is continuous at $x = 0$.</p><p>∴ Answer is (b).</p>
Correct Answer: b