The correct option is B 0 lim_x →∞{√(x^2+√(x^4+1)) √(2)x} lim_x →∞ #160;√(x^4+1) x^2/(√(x^2+√(x^4+1))+√(2x)) lim_x →∞1/(√(x^2+√(x^4+1))+√ ...

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Rationalization Method to Remove Indeterminate Form

limx →∞√x2+√x...

Question

$lim x \to \infty {\sqrt{x^{2} + \sqrt{x^{4} + 1}} - \sqrt{2} x}$

\infty

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Solution

The correct option is B $0$
$lim x \to \infty {\sqrt{x^{2} + \sqrt{x^{4} + 1}} - \sqrt{2} x}$
$lim x \to \infty \frac{\sqrt{x^{4} + 1} - x^{2}}{(\sqrt{x^{2} + \sqrt{x^{4} + 1}} + \sqrt{2 x})}$
$lim x \to \infty \frac{1}{(\sqrt{x^{2} + \sqrt{x^{4} + 1}} + \sqrt{2 x}) (\sqrt{x^{4} + 1} + x^{2})}$
$lim x \to \infty \frac{1}{x^{3} (\sqrt{1 + \sqrt{1 + \frac{1}{x^{4}}}} + \sqrt{2}) (\sqrt{1 + \frac{1}{x^{4}}} + 1)}$
$= 0$

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