lim x ( 3 x + 1 + 3 x - 1 ) 6 + ( 3 x + 1 - 3 x - 1 ) 6 ( x + x 2 - 1 ) 6 + ( x - x 2 - 1 ) 6 x 3 [2023]

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Solution

Q.
$\lim_{x \to \infty} \frac{{(\sqrt{3 x + 1} + \sqrt{3 x - 1})}^{6} + {(\sqrt{3 x + 1} - \sqrt{3 x - 1})}^{6}}{{(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}} x^{3}$ [2023]

1 is equal to $\frac{27}{2}$

2 is equal to 9

3 does not exist

4 is equal to 27

Ans.
(4)

$\lim_{x \to \infty} \frac{{(\sqrt{3 x + 1} + \sqrt{3 x - 1})}^{6} + {(\sqrt{3 x + 1} - \sqrt{3 x - 1})}^{6}}{{(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}} \times x^{3}$

$\lim_{x \to \infty} x^{3} \times {\frac{x^{3} [{(\sqrt{3 + \frac{1}{x}} + \sqrt{3 - \frac{1}{x}})}^{6} + {(\sqrt{3 + \frac{1}{x}} - \sqrt{3 - \frac{1}{x}})}^{6}]}{x^{6} [{(1 + \sqrt{1 - \frac{1}{x^{2}}})}^{6} + {(1 - \sqrt{1 - \frac{1}{x^{2}}})}^{6}]}}$

$= \frac{{(\sqrt{3} + \sqrt{3})}^{6} + {(\sqrt{3} - \sqrt{3})}^{6}}{{(1 + \sqrt{1})}^{6} + {(1 - \sqrt{1})}^{6}} = \frac{{(2 \sqrt{3})}^{6} + 0}{2^{6} + 0} = \frac{2^{6} \cdot 3^{3}}{2^{6}} = 3^{3} = 27$

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