# How do you find the limit of ( 1 - (5/x) ) ^x as x approaches infinity?

Jun 7, 2016

the limit is : ${e}^{-} 5$

#### Explanation:

the limit is of the form 1^∞

${\lim}_{x \to a} {\left(f \left(x\right)\right)}^{g \left(x\right)} = {\lim}_{x \to a} {\left(1 + \left(f \left(x\right) - 1\right)\right)}^{g \left(x\right)}$

 lim_(x->a) (f(x))^(g(x)) = e^(lim_(x->a) ( f(x) - 1 ) * (g(x))

hence,
In the question given, the limit is of the form : ${e}^{k}$

$k = {\lim}_{x \to 0} \left(- \frac{5}{x}\right) \cdot \left(x\right)$

$k = - 5$

Therefore, the limit of the above function is : ${e}^{- 5}$