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

A
$e^{1 / 2}$
B
$e^{3 / 2}$
C
$e^{3}$
D
none of these

Solution:

We have,

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

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