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Question
\[\lim_{x \to 0} \frac{2x}{\sqrt{a + x} - \sqrt{a - x}}\]
Solution
\[\lim_{x \to 0} \left[ \frac{2x}{\sqrt{a + x} - \sqrt{a - x}} \right]\]
When x = 0, then the expression \[\frac{2x}{\sqrt{a + x} - \sqrt{a - x}}\] becomes \[\frac{0}{0}\] Rationalising the denominator:
\[\lim_{x \to 0} \left[ \frac{2x}{\left( \sqrt{a + x} - \sqrt{a - x} \right)} \times \frac{\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( \sqrt{a + x} + \sqrt{a - x} \right)} \right]\]
\[\lim_{x \to 0} \left[ \frac{\left( 2x \right)\left( \sqrt{a + x} + \sqrt{a - x} \right)}{\left( a + x \right) - \left( a - x \right)} \right]\]
\[\lim_{x \to 0} \left[ \frac{2x\left( \sqrt{a + x} + \sqrt{a - x} \right)}{2x} \right]\]
= \[\sqrt{a} + \sqrt{a}\]
\[2\sqrt{a}\]
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