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Question
\[\lim_{x \to 5} \frac{x - 5}{\sqrt{6x - 5} - \sqrt{4x + 5}}\]
Solution
\[\lim_{x \to 5} \left[ \frac{x - 5}{\sqrt{6x - 5} - \sqrt{4x + 5}} \right]\] It is of the form \[\frac{0}{0}\]
Rationalising the denominator:
\[\lim_{x \to 5} \left[ \frac{\left( x - 5 \right) \left( \sqrt{6x - 5} + \sqrt{4x + 5} \right)}{\left( \sqrt{6x - 5} - \sqrt{4x + 5} \right) \left( \sqrt{6x - 5} + \sqrt{4x + 5} \right)} \right]\]
= \[\lim_{x \to 5} \left[ \frac{\left( x - 5 \right) \left( \sqrt{6x - 5} + \sqrt{4x + 5} \right)}{\left( 6x - 5 \right) - \left( 4x + 5 \right)} \right]\]
= \[\lim_{x \to 5} \left[ \frac{\left( x - 5 \right) \left( \sqrt{6x - 5} + \sqrt{4x + 5} \right)}{2x - 10} \right]\]
=\[\lim_{x \to 5} \left[ \frac{\left( x - 5 \right)\left( \sqrt{6x - 5} + \sqrt{4x + 5} \right)}{2\left( x - 5 \right)} \right]\]
= \[\frac{\sqrt{6 \times 5 - 5} + \sqrt{4 \times 5 + 5}}{2}\]
= \[\frac{5 + 5}{2}\]
= 5
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