lim _ x→∞ #160; ( √( 2 x ^ 2 3 ) 5x ) #160; =lim _ x→∞ #160; ( √( 2 x ^ 2 3 ) 5x ) ( √( 2 x ^ 2 3 ) +5x ) #160; √( 2 x ^ 2 3 ) +5x #1 ...

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Rationalization Method to Remove Indeterminate Form

limx→∞ √2x2-3...

Question

$lim x \to \infty (\sqrt{2 x^{2} - 3} - 5 x)$

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Solution

${lim}_{x \to \infty} (\sqrt{2 x^{2} - 3} - 5 x)$
$= {lim}_{x \to \infty} \frac{(\sqrt{2 x^{2} - 3} - 5 x) (\sqrt{2 x^{2} - 3} + 5 x)}{\sqrt{2 x^{2} - 3} + 5 x}$
$= {lim}_{x \to \infty} \frac{2 x^{2} - 3 - 25 x^{2}}{x \sqrt{2 - \frac{3}{x^{2}}} + 5 x}$
$= {lim}_{x \to \infty} \frac{(- 22 - \frac{3}{x^{2}}) x^{2}}{x (5 + \sqrt{2 - \frac{3}{x^{2}}})}$
$= x {lim}_{x \to \infty} \frac{1 (- 22)}{5 + \sqrt{2}} = \infty$

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