The correct option is C 1 As x^ 3/2 (√( x ^ 3 +1 ) √( x ^ 3 1 ) )= x ^ 3/2 ( x ^ 3 +1 ) ( x ^ 3 1 ) nbsp;( √( x ^ 3 +1 ) +√( x ^ 3 1 ) nb ...

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Rationalization Method to Remove Indeterminate Form

limx→∞x3/2√x3...

Question

$lim x \to \infty x^{3 / 2} (\sqrt{x^{3} + 1} - \sqrt{x^{3} - 1}) =$

- 1

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\frac{1}{2}

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1

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Solution

The correct option is C $1$
As $x^{3 / 2} (\sqrt{x^{3} + 1} - \sqrt{x^{3} - 1}) = x^{3 / 2} \frac{(x^{3} + 1) - (x^{3} - 1)}{(\sqrt{x^{3} + 1} + \sqrt{x^{3} - 1})} = \frac{{2 x}^{3 / 2}}{(\sqrt{x^{3} + 1} + \sqrt{x^{3} - 1})} = \frac{2}{(\sqrt{1 + \frac{1}{x^{3}}} + \sqrt{1 - \frac{1}{x^{3}}})}$
Then
${lim}_{x \to \infty} x^{3 / 2} (\sqrt{x^{3} + 1} - \sqrt{x^{3} - 1}) = {lim}_{x \to \infty} \frac{2}{(\sqrt{1 + \frac{1}{x^{3}}} + \sqrt{1 - \frac{1}{x^{3}}})} = 1$
Hence, option 'D' is correct.

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