Limits, Continuity & Differentiability
Continuity at a Point
Grade 12
Question:
<p>If function <span class="math">f(x) = \frac{\sqrt{1+x}-\sqrt{1-x}}{x}\cdot\frac{1}{3}\sqrt{1+x}\cdot\frac{1}{x}\cdot\sqrt{1}\cdot x</span> is continuous function at <span class="math">x=0</span>, then <span class="math">f(0)</span> is equal to</p>
<p>(a) <span class="math">\frac{1}{2}</span></p>
<p>(b) <span class="math">\frac{1}{4}</span></p>
<p>(c) <span class="math">\frac{1}{6}</span></p>
<p>(d) <span class="math">\frac{1}{3}</span></p>
Step-by-Step Solution
Key Concept: Apply continuity condition and L'Hôpital's rule to evaluate the limit as x approaches 0.
<p><strong>Solution:</strong> For continuity at <span class="math">x=0</span>, we need <span class="math">\lim_{x \to 0} f(x) = f(0)</span>. Using L'Hôpital's rule or algebraic manipulation: <span class="math">\lim_{x \to 0} \frac{\sqrt{1+x}-\sqrt{1-x}}{x} = \lim_{x \to 0} \frac{\frac{1}{2\sqrt{1+x}} + \frac{1}{2\sqrt{1-x}}}{1} = \frac{1}{2}</span>. After simplification with the remaining terms, <span class="math">f(0) = \frac{1}{4}</span>.</p>
Correct Answer: b