`("lim")_(n vec oo)"{"(n/(n+1))^(alpha)+sin (1/n)]^n`(when `alpha in Q`) is equal to (a) `e^(-alpha)` (b) `-alpha` (c) `e^(1-alpha)` (d) `e^(1+alpha)`

Options

A. `e^alpha`
B. `-alpha`
C. `e^1-alpha`
D. `e^1+alpha`

Answer

C

Explanation

`lim_(nto oo) {((n)/(n+1))^alpha+sin.(1)/(n)}^n`
`=lim_(nto oo) {(1+(n)/(n))^alpha+sin.(1)/(n)}^n`
`lim_(nto oo) {1+(n)/n+(alpha(alpha+ 1)) /(2!).(1)/(n^2) ....}+{(1)/(n)-(1)/(3!n^3)+(1)/(5!n^5)-....}}^n`
`=lim_(nto oo) {1+((1-alpha)/(n))+(alpha(alpha+ 1)) /(2!n^2). ....}^n`
`=lim_(nto oo) {((1-alpha)/(n))+(alpha(alpha+ 1)) /(2!n^2)+ ....}n`
`=e ^(lim_(xtooo) {((1-alpha)/(n))+(alpha(alpha+))/(2!n^2)+...}n)=e^(1-alpha)`