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Question
\[\lim_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^2 - 1}\]
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Solution
\[\lim_{x \to 1} \left[ \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^2 - 1} \right]\] It is of the form \[\frac{0}{0}\]
Rationalising the numerator:
\[\lim_{x \to 1} \left[ \frac{\left( \sqrt{3 + x} - \sqrt{5 - x} \right) \left( \sqrt{3 + x} + \sqrt{5 - x} \right)}{\left( x - 1 \right) \left( x + 1 \right) \left( \sqrt{3 + x} + \sqrt{5 - x} \right)} \right]\]
\[ = \lim_{x \to 1} \left[ \frac{\left( 3 + x \right) - \left( 5 - x \right)}{\left( x - 1 \right) \left( x + 1 \right) \left( \sqrt{3 + x} + \sqrt{5 - x} \right)} \right]\]
\[ = \lim_{x \to 1} \frac{2\left( x - 1 \right)}{\left( x - 1 \right) \left( x + 1 \right) \left( \sqrt{3 + x} + \sqrt{5 - x} \right)}\]
\[ = \frac{2}{\left( 1 + 1 \right) \left( \sqrt{3 + 1} + \sqrt{5 - 1} \right)}\]
\[ = \frac{2}{2 \times \left( 2 + 2 \right)}\]
\[ = \frac{1}{4}\]
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