is equal to : [2025]

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Solution

Q.
$\lim_{x \to \infty} \frac{(2 x^{2} - 3 x + 5) {(3 x - 1)}^{\frac{x}{2}}}{(3 x^{2} + 5 x + 4) \sqrt{{(3 x + 2)}^{x}}}$ is equal to : [2025]

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

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

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

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

Ans.
(2)

$\lim_{x \to \infty} \frac{(2 - \frac{3}{x} + \frac{5}{x^{2}}) {(1 - \frac{1}{3 x})}^{x / 2}}{(3 + \frac{5}{x} + \frac{4}{x^{2}}) {(1 + \frac{2}{3 x})}^{x / 2}}$

$= \lim_{x \to \infty} \frac{2}{3} \cdot \frac{e^{\frac{x}{2} (1 - \frac{1}{3 x} - 1)}}{e^{\frac{x}{2} (1 + \frac{2}{3 x} - 1)}}$

$= \frac{2}{3} \cdot \frac{e^{- \frac{1}{6}}}{e^{\frac{1}{3}}} = \frac{2}{3 \sqrt{e}}$ .

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