The value of lim x 1 + 2 - 3 + 4 + 5 - 6 + … + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n 2 n 4 + 4 n + 3 - n 4 + 5 n + 4 is [2023]

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Solution

Q.
$The value of \lim_{x \to \infty} \frac{1 + 2 - 3 + 4 + 5 - 6 + \dots + (3 n - 2) + (3 n - 1) - 3 n}{\sqrt{2 n^{4} + 4 n + 3} - \sqrt{n^{4} + 5 n + 4}} is$ [2023]

1 $3 (\sqrt{2} + 1)$

2 $\frac{3}{2} (\sqrt{2} + 1)$

3 $\frac{\sqrt{2} + 1}{2}$

4 $\frac{3}{2 \sqrt{2}}$

Ans.
(2)

$\lim_{n \to \infty} \frac{1 + 2 - 3 + 4 + 5 - 6 + \dots + (3 n - 2) + (3 n - 1) - 3 n}{\sqrt{2 n^{4} + 4 n + 3} - \sqrt{n^{4} + 5 n + 4}}$

$= \lim_{n \to \infty} \frac{(1 + 2 + 3 + 4 + \dots + 3 n) - 2 (3 + 6 + 9 + \dots + 3 n)}{\sqrt{2 n^{4} + 4 n + 3} - \sqrt{n^{4} + 5 n + 4}}$

$= \lim_{n \to \infty} \frac{[\frac{3 n (3 n + 1)}{2} - \frac{6 n (n + 1)}{2}] [\sqrt{2 n^{4} + 4 n + 3} + \sqrt{n^{4} + 5 n + 3}]}{(n^{4} - n - 1)}$

$= \lim_{n \to \infty} \frac{(3 n^{2} - 3 n) \times n^{2} \times (\sqrt{2 + \frac{4}{n^{3}} + \frac{3}{n^{4}}} + \sqrt{1 + \frac{5}{n^{3}} + \frac{4}{n^{4}}})}{2 (n^{4} - n - 1)}$

$= \lim_{n \to \infty} \frac{(3 - \frac{3}{n}) (\sqrt{2 + \frac{4}{n^{3}} + \frac{3}{n^{4}}} + \sqrt{1 + \frac{5}{n^{3}} + \frac{4}{n^{4}}})}{2 (1 - \frac{1}{n^{3}} - \frac{1}{n^{4}})}$

$= \frac{3 (\sqrt{2} + 1)}{2}$

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