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Question
\[\lim_{x \to 0} \frac{\sqrt{a + x} - \sqrt{a}}{x\sqrt{a^2 + ax}}\]
Solution
\[\lim_{x \to 0} \left[ \frac{\sqrt{a + x} - \sqrt{a}}{x\sqrt{a^2 + ax}} \right]\] It is of the form \[\frac{0}{0}\]
Rationalising the numerator:
\[\lim_{x \to 0} \left[ \frac{\left( \sqrt{a + x} - \sqrt{a} \right) \times \left( \sqrt{a + x} + \sqrt{a} \right)}{\left( x\sqrt{a^2 + ax} \right) \times \left( \sqrt{a + x} + \sqrt{a} \right)} \right]\]
= \[\lim_{x \to 0} \left[ \frac{a + x - a}{x\left( \sqrt{a^2 + ax} \right) \left( \sqrt{a + x} + \sqrt{a} \right)} \right]\]
= \[\frac{1}{\left( \sqrt{a^2} \right) \left( \sqrt{a} + \sqrt{a} \right)}\]
= \[\frac{1}{2a\sqrt{a}}\]
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