Let m and n be two positive integers greater then 1. If ` lim_( a to 0) ((e^(cos(alpha^(n)))-e/alpha^(m)) =- (e/2) ` , then the value of `m/n` is

Let m and n be two positive integers greater then 1. If ` lim_( a to 0

1.	Let m and n be two positive integers greater then 1. If ` lim_( a to 0) ((e^(cos(alpha^(n)))-e/alpha^(m)) =- (e/2) ` , then the value of `m/n` is
Answer» Correct Answer - `(2)` Given, `underset( alpha to 0) lim [(e^(cos(alpha^(n))-e))/alpha^(m)]=- e/2` `rArr underset( alphato 0)lim(e{e^(cos(alpha^(n))-1)})/(cos (alpha^(n))-1)(cos(alpha^(n))-1)/alpha^(m) = (-e)/2` `rArr underset( alpha to 0) lim e{(e^(cos(alpha^(n))-1))/(cos(alpha^(n))-1)}underset(alphato 0) lim (-2sin^(2) .alpha^(n)/2)/alpha^(m) =- e//2` `rArr e xx1 (-2) underset( alphato 0) lim (sin^(2).((alpha^(n))/2))/(alpha^(2n)/4 )* alpha^(2n)/(4 alpha^(m)) = (-e)/2` ` rArr e xx 1xx -2 xx1 underset( alphato 0) lim (alpha^(2n-m))/4 = (-e)/2 ` For this to be exists, 2n - m = 0 ` rArr" " m/n = 2`