Value of `lim_(a to oo) (e.a^(2).e^(3).a^(4). ..........e^(n-1)a^(n))^((1)/(n^(2)+1))`is

Options

A. 1
B. 0
C. `sqrt(ab)`
D. `4sqrt(ae)`

Answer

D

Explanation

it is obviious n is even then
`underset(n to oo)lim (e^(1+3+5......(n)/(2)"terms"),a^(2+4+6+.......(n)/(2)"terms"))^((1)/(n^(2)+1))`
`implies underset(n to oo)lim (e^((n^(2))/(4)).a^((n(n+2))/(4)))^((1)/(n^(2)+1))=underset(n to oo) lim e^((n^(2))/(4(n+1))).a^((n(n+2))/(4(n^(2)+1)))`
`=e ^(underset(n to oo)lim(1)/(4(l+(1//n^(2))))).a^( underset(a to oo)lim ((1+(2//n)))/(4(1+(l//n^(2)))))`
`=e^((1)/(4)).a^((1)/(4))`
`=(e.a)^((1)/(4))`