The value of limx→∞ x2+2x+32×2+x+53x−23x+2 , is

# The value of $\underset{x\to \mathrm{\infty }}{lim} {\left(\frac{{x}^{2}+2x+3}{2{x}^{2}+x+5}\right)}^{\frac{3x-2}{3x+2}}$ , is

1. A

${e}^{1/2}$

2. B

${e}^{3/2}$

3. C

${e}^{3}$

4. D

none of these

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### Solution:

We have,

$\underset{x\to \mathrm{\infty }}{lim} {\left(\frac{{x}^{2}+2x+3}{2{x}^{2}+x+5}\right)}^{\frac{3x-2}{3x+2}}=\underset{x\to \mathrm{\infty }}{lim} {\left(\frac{1+\frac{2}{x}+\frac{3}{{x}^{2}}}{2+\frac{1}{x}+\frac{5}{{x}^{2}}}\right)}^{\frac{1-2/3x}{1+2/3x}}={\left(\frac{1}{2}\right)}^{1}=\frac{1}{2}$

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