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