Answer 1

\[\lim_{x \to 1} \left[ \frac{\sqrt{5x - 4} - \sqrt{x}}{x^3 - 1} \right]\] It is of the form \[\frac{0}{0}\] 

Rationalising the numerator: 

\[\lim_{x \to 1} \left[ \frac{\left( \sqrt{5x - 4} - \sqrt{x} \right) \left( \sqrt{5x - 4} + \sqrt{x} \right)}{\left( x^3 - 1 \right) \left( \sqrt{5x - 4} + \sqrt{x} \right)} \right]\] 

= \[\lim_{x \to 1} \left[ \frac{5x - 4 - x}{\left( x - 1 \right)\left( x^2 + x + 1 \right)\left( \sqrt{5x - 4} + \sqrt{x} \right)} \right]\] 

=  \[\lim_{x \to 1} \left[ \frac{4\left( x - 1 \right)}{\left( x - 1 \right)\left( x^2 + x + 1 \right)\left( \sqrt{5x - 4} + \sqrt{x} \right)} \right]\] 

=  \[\frac{4}{3\left( 1 + 1 \right)}\] 

= \[\frac{2}{3}\]