lim_x→ 1√(5x 4) √(x)x^3 1×√(5x 4)+√(x)√(5x 4)+√(x) (Rationalising) =lim_x→ 1(5x 4) x(x^3 1)(x√(5x 4)+√(x)) =lim_x→ 14(x 1)(x^3 1)(√(5x 4)+√(x)) ...

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Property 1

limx→ 1√5x-4-...

Question

${lim}_{x \to 1} \frac{\sqrt{5 x - 4} - \sqrt{x}}{x^{3} - 1}$

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Solution

$lim x \to 1 \frac{\sqrt{5 x - 4} - \sqrt{x}}{x^{3} - 1} \times \frac{\sqrt{5 x - 4} + \sqrt{x}}{\sqrt{5 x - 4} + \sqrt{x}}$ (Rationalising)
$= lim x \to 1 \frac{(5 x - 4) - x}{(x^{3} - 1) (x \sqrt{5 x - 4} + \sqrt{x})}$
$= lim x \to 1 \frac{4 (x - 1)}{(x^{3} - 1) (\sqrt{5 x - 4} + \sqrt{x})}$
$= lim x \to 1 \frac{4 (x - 1)}{(x - 1) (x^{2} + x + 1) (\sqrt{5 x - 4} + \sqrt{x})}$
$= lim x \to 1 \frac{4}{(x^{2} + x + 1) (\sqrt{5 x - 4} + \sqrt{x})}$
$= \frac{4}{(1^{2} + 1 + 1) (\sqrt{5 - 4} + \sqrt{1})}$
$= \frac{4}{3 (2)}$
$= \frac{2}{3}$ .

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