_ x , ( x + a x + b )^ x + b = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to \infty } \,{\left( {\frac{{x + a}}{{x + b}}} \right)^{x + b}} = $

A
$1$
B
${e^{b - a}}$
✓
${e^{a - b}}$
D
${e^b}$

✓

Answer

Correct option: C.

${e^{a - b}}$

c
(c) $\mathop {{\rm{lim}}}\limits_{x \to \infty } \,{\left( {\frac{{x + a}}{{x + b}}} \right)^{x + b}} = \mathop {{\rm{lim}}}\limits_{x \to \infty } \,{\left( {1 + \frac{{a - b}}{{x + b}}} \right)^{x + b}}$

$ = \mathop {{\rm{lim}}}\limits_{x \to \infty } \,{\left\{ {{{\left( {1 + \frac{{a - b}}{{x + b}}} \right)}^{\frac{{x + b}}{{a - b}}}}} \right\}^{a - b}} = {e^{a - b}}$.

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