_ x , ( x^2 + 5x + 3 x^2 + x + 3 )^x = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to \infty } \,{\left( {\frac{{{x^2} + 5x + 3}}{{{x^2} + x + 3}}} \right)^x}$=

✓
${e^4}$
B
${e^2}$
C
${e^3}$
D
$e$

✓

Answer

Correct option: A.

${e^4}$

a
(a) $\frac{{{x^2} + 5x + 3}}{{{x^2} + x + 3}} = 1 + \frac{{4x}}{{{x^2} + x + 3}} = 1 + y$ (say)
where $y = \frac{{4x}}{{{x^2} + x + 3}} = \frac{{\frac{4}{x}}}{{1 + \frac{1}{x} + \frac{3}{{{x^2}}}}} = 0$ as $x \to \infty $
Also, $xy = \frac{{4{x^2}}}{{{x^2} + x + 3}} = \frac{4}{{1 + \frac{1}{x} + \frac{3}{{{x^2}}}}} = 4$ as $x \to \infty $
$\therefore$ limit $ = \mathop {\lim }\limits_{y \to 0} {(1 + y)^x}$ $ = \mathop {\lim }\limits_{y \to 0} {[{(1 + y)^{1/y}}]^{\,xy}} = {e^{xy}} = {e^4}$.

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