If `f(x)=lim_(nrarroo) (cos(x)/(sqrtn))^(n)`, then the value of `lim_(xrarr0) (f(x)-1)/(x)` is

Options

A. 0
B. 1
C. 2
D. `3//2`

Answer

A

Explanation

`f(x)=underset(nrarroo)(lim)(cos.(x)/(sqrtn))^(n)`
`" "=e^(underset(nrarroo)(lim)(cos.(x)/(sqrtn)-1).n)`
`" "e^(-(2underset(nrarroo)(lim)(x^(2))/(4)(sin^(2).(x)/(2sqrtn))/((x^(2))/(4sqrtn))))`
`" "=e^(-(1)/(2)x^(2))`
`rArr" "underset(xrarr0)(lim)(f(x)-1)/(x)=underset(xrarr0)(lim)(e^(-(1)/(2)x^(2))-1)/(x)=0`