If `f(n)=underset(xto0)lim{(1+"sin"(x)/(2))(1+"sin"(x)/(2^(2)))...(1+"sin"(x)/(2^(n)))}^((1)/(x))` then find `underset(ntooo)limf(n).`

Options

A. 1
B. e
C. 0
D. `oo`

Answer

B

Explanation

`f(n)=underset(xrarr0)(lim)e^((1)/(x)((1+sin.(x)/(2))(1+sin.(x)/(2^(2)))...(1+sin.(x)/(2^(n)))-1))`
`=underset(xrarr0)(lim)e^(((1+(sin.(x)/(2)+sin.(x)/(2^(2))+...+sin.(x)/(2^(n)))+(sin.(x)/(2)sin.(x)/(2^(2))+...)+...-1))/(x))`
`=underset(xrarr0)(lim)e^(((sin.(x)/(2))/(x)+(sin((x)/(2^(2))))/(2)+...+sin.(((x)/(2^(n))))/(x)))`
`=e^(((1)/(2)+(1)/(2^(2))+...(1)/(2^(n))))`
`therefore underset(nrarroo)(lim)f(n)=e^((1//2)/(1-(1)/(2)))=e`