_ x ( x + 3 x + 1 )^ x + 1 = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{x + 3}}{{x + 1}}} \right)^{x + 1}} = $

✓
${e^2}$
B
${e^3}$
C
$e$
D
${e^{ - 1}}$

✓

Answer

Correct option: A.

${e^2}$

a
(a) $\mathop {\lim }\limits_{x \to \infty } {\left( {\frac{{x + 3}}{{x + 1}}} \right)^{x + 1}}$

$ = \mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{2}{{x + 1}}} \right)^{\frac{{x + 1}}{2}.2}}$

$ = {\left\{ {\mathop {\lim }\limits_{x \to \infty } {{\left( {1 + \frac{2}{{x + 1}}} \right)}^{\frac{{x + 1}}{2}}}} \right\}^2}$$ = {e^2}$.

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