Question
For $$x \in R,\,\,\,\mathop {\lim }\limits_{x\, \to \,\infty } {\left( {\frac{{x - 3}}{{x + 2}}} \right)^x} = ?$$
A.
$$e$$
B.
$${e^{ - \,1}}$$
C.
$${e^{ - \,5}}$$
D.
$${e^{ 5}}$$
Answer :
$${e^{ - \,5}}$$
Solution :
$$\eqalign{
& {\text{For }}x \in R, \cr
& \mathop {\lim }\limits_{x\, \to \,\infty } {\left( {\frac{{x - 3}}{{x + 2}}} \right)^x} = \mathop {\lim }\limits_{x\, \to \,\infty } {\left\{ {{{\left[ {1 - \frac{5}{{x + 2}}} \right]}^{\frac{{ - \,\left( {x + 2} \right)}}{5}}}} \right\}^{\frac{{ - \,5x}}{{x + 2}}}} \cr
& = {e^{\mathop {\lim }\limits_{x \to \,\infty } - \,\frac{5}{{1\, + \,\frac{2}{x}}}}} = {e^{ - \,5}} \cr} $$