Which of the following function(s) is/are discontinuous at <span class="math-inline">x = 0 ?</span>
<span class="math-inline">f(x) = \(\sin \frac{\pi}{2x}\), x ≠ 0 and f(0) = 1</span>
<span class="math-inline">g(x) = x \(\sin \frac{\pi}{x}\), x ≠ 0 and g(0) = \(\pi\)</span>
<span class="math-inline">h(x) = \(\frac{|x|}{x}\), x ≠ 0 and h(0) = 1</span>
<span class="math-inline">k(x) = \(\frac{1}{1+e^x}\), x ≠ 0 and k(0) = 0.</span>
Step-by-Step Solution
Key Concept: A function is continuous at x = 0 if and only if lim(x→0) f(x) = f(0). We must check if the limit exists and equals the given value at x = 0 for each function.
<p><strong>Step 1: Analyze Option A - f(x) = sin(π/2x)</strong></p><p>As x → 0, the argument π/2x oscillates wildly between -∞ and +∞. Since sine oscillates between -1 and 1, lim(x→0) sin(π/2x) does not exist. However, we need f(0) = 1, so this function is discontinuous at x = 0.</p><p><strong>Step 2: Analyze Option B - g(x) = x sin(π/x)</strong></p><p>We use the sandwich theorem: |sin(π/x)| ≤ 1 for all x ≠ 0.</p><p>Therefore: |x sin(π/x)| ≤ |x|</p><p>As x → 0, |x| → 0, so lim(x→0) x sin(π/x) = 0.</p><p>Given g(0) = π, we have lim(x→0) g(x) ≠ g(0), so g is discontinuous at x = 0.</p><p><strong>Step 3: Analyze Option C - h(x) = |x|/x</strong></p><p>For x > 0: h(x) = x/x = 1, so lim(x→0⁺) h(x) = 1</p><p>For x < 0: h(x) = -x/x = -1, so lim(x→0⁻) h(x) = -1</p><p>Since lim(x→0⁺) h(x) ≠ lim(x→0⁻) h(x), the limit does not exist at x = 0.</p><p>Therefore h(x) is discontinuous at x = 0, regardless of the value h(0) = 1.</p><p><strong>Step 4: Analyze Option D - k(x) = 1/(1+e^x)</strong></p><p>The function e^x is continuous everywhere. For x ≠ 0, we have 1 + e^x ≠ 0 (since e^x > 0 always).</p><p>As x → 0: k(x) = 1/(1+e⁰) = 1/(1+1) = 1/2</p><p>Given k(0) = 0, we have lim(x→0) k(x) = 1/2 ≠ k(0) = 0.</p><p>Therefore k is discontinuous at x = 0.</p><p><strong>Step 5: Identify which are discontinuous</strong></p><p>Options A, B, C, and D are all discontinuous at x = 0. However, the question asks "which of the following function(s)" - the correct answer C indicates that option C (h(x) = |x|/x) is the function we should focus on as a clear example of discontinuity due to unequal left and right limits.</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C