Question

lim x—1 (√5x-4 -√x)/x³-1​

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \sf{\lim_{x \to 1} \: \frac{ \sqrt{5x - 4} - \sqrt{x} }{ {x}^{3} - 1}}$

EVALUATION

$\displaystyle \sf{\lim_{x \to 1} \: \frac{ \sqrt{5x - 4} - \sqrt{x} }{ {x}^{3} - 1}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ (\sqrt{5x - 4} + \sqrt{x} )( \sqrt{5x - 4} - \sqrt{x} )}{ ({x}^{3} - 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ (5x - 4 - x)}{ (x - 1)( {x}^{2} + x + 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ (4x - 4 )}{ (x - 1)( {x}^{2} + x + 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ 4 }{( {x}^{2} + x + 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \: \frac{ 4 }{( {1}^{2} + 1 + 1)( \sqrt{5 - 4} + \sqrt{1} )}}$

$\: \displaystyle \sf{ = \: \frac{ 4 }{3 \times 2}}$

$\: \displaystyle \sf{ = \: \frac{2}{3 }}$

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Answer 2

Solution ⤵️

To Find ⤵️

$\displaystyle \sf{\lim_{x \to 1} \: \frac{ \sqrt{5x - 4} - \sqrt{x} }{ {x}^{3} - 1}}$

Calculation ⤵️

$\displaystyle \sf{\lim_{x \to 1} \: \frac{ \sqrt{5x - 4} - \sqrt{x} }{ {x}^{3} - 1}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ (\sqrt{5x - 4} + \sqrt{x} )( \sqrt{5x - 4} - \sqrt{x} )}{ ({x}^{3} - 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ (5x - 4 - x)}{ (x - 1)( {x}^{2} + x + 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ (4x - 4 )}{ (x - 1)( {x}^{2} + x + 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \lim_{x \to 1} \: \frac{ 4 }{( {x}^{2} + x + 1)( \sqrt{5x - 4} + \sqrt{x} )}}$

$\: \displaystyle \sf{ = \: \frac{ 4 }{( {1}^{2} + 1 + 1)( \sqrt{5 - 4} + \sqrt{1} )}}$

$\: \displaystyle \sf{ = \: \frac{ 4 }{3 \times 2}}$

$\: \displaystyle \sf{ = \: \frac{2}{3 }}$

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