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Question
\[\lim_{x \to \sqrt{10}} \frac{\sqrt{7 + 2x} - \left( \sqrt{5} + \sqrt{2} \right)}{x^2 - 10}\]
Solution
\[\lim_{x \to \sqrt{10}} \left[ \frac{\sqrt{7 + 2x} - \left( \sqrt{5} + \sqrt{2} \right)}{x^2 - \left( \sqrt{10} \right)^2} \right]\]
= \[\lim_{x \to \sqrt{10}} \left[ \frac{\sqrt{7 + 2x} - \sqrt{\left( \sqrt{5} + \sqrt{2} \right)^2}}{\left( x - \sqrt{10} \right)\left( x + \sqrt{10} \right)} \right]\]
= \[\lim_{x \to \sqrt{10}} \left[ \frac{\sqrt{7 + 2x} - \sqrt{5 + 2 + 2\sqrt{5}\sqrt{2}}}{\left( x - \sqrt{10} \right)\left( x + \sqrt{10} \right)} \right]\]
= \[\lim_{x \to \sqrt{10}} \left[ \frac{\sqrt{7 + 2x} - \sqrt{7 + 2\sqrt{10}}}{\left( x - \sqrt{10} \right)\left( x + \sqrt{10} \right)} \right]\]
Rationalising the numerator:
\[\lim_{x \to \sqrt{10}} \left[ \frac{\left( \sqrt{7 + 2x} - \sqrt{7 + 2\sqrt{10}} \right) \left( \sqrt{7 + 2x} + \sqrt{7 + 2\sqrt{10}} \right)}{\left( x - \sqrt{10} \right)\left( x + \sqrt{10} \right) \left( \sqrt{7 + 2x} + \sqrt{7 + 2\sqrt{10}} \right)} \right]\]
= \[\lim_{x \to \sqrt{10}} \left[ \frac{\left( 7 + 2x \right) - \left( 7 + 2\sqrt{10} \right)}{\left( x - \sqrt{10} \right)\left( x + \sqrt{10} \right) \left( \sqrt{7 + 2x} + \sqrt{7 + 2\sqrt{10}} \right)} \right]\]
= \[\lim_{x \to \sqrt{10}} \left[ \frac{2\left( x - \sqrt{10} \right)}{\left( x - \sqrt{10} \right)\left( x + \sqrt{10} \right) \left( \sqrt{7 + 2x} + \sqrt{7 + 2\sqrt{10}} \right)} \right]\]
= \[\frac{2}{\left( \sqrt{10} + \sqrt{10} \right) \left( 2\sqrt{7 + 2\sqrt{10}} \right)}\]
\[= \frac{2}{\left( 2\sqrt{10} \right) \times 2\sqrt{7 + 2\sqrt{10}}}\]
\[ = \frac{1}{2\sqrt{10}\sqrt{\left( \sqrt{5} + \sqrt{2} \right)^2}}\]
= \[\frac{1}{2\sqrt{10}\left( \sqrt{5} + \sqrt{2} \right)} \times \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} - \sqrt{2}}\]
\[= \frac{1}{2\sqrt{10}}\left[ \frac{\sqrt{5} - \sqrt{2}}{\left( \sqrt{5} \right)^2 - \left( \sqrt{2} \right)^2} \right]\]
= \[\frac{\sqrt{5} - \sqrt{2}}{6\sqrt{10}}\]
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