Question

value of lim(x-->infinite)(1/x-1/x²+1/x³... infinite)^1/x is equal to​

Answer 1

Answer:

(D) 1

Step-by-step explanation:

We have,

$\lim_{x \rarr \infty }( \frac{1}{x} - \frac{1}{ {x}^{2} } + \frac{1}{ {x}^{3} } - ...) ^{ \frac{1}{x} }$

$\lim_{x \rarr \infty }( \frac{ \frac{1}{x} }{1 - \frac{1}{x} } ) ^{ \frac{1}{x} }$

$= \lim_{x \rarr \infty }( \frac{1}{(x - 1)} )^{ \frac{1}{x} }$

$= \lim_{x \rarr \infty } {e}^{ \frac{1}{x} ln( \frac{1}{(x - 1)} )}$

$= \lim_{x \rarr \infty } {e}^{ - \frac{ ln(x - 1) }{x} }$

Using l'hospital rule,

$= \lim_{x \rarr \infty } {e}^{ - \frac{1 }{x - 1} }$

$= {e}^{0} = 1$