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