If `lim_(x rarr 0)(e^(-nx)+e^(nx)-2 cos\ (nx)/2 - kx^2)/(sin x - tan x)` exists and finite `(n,k in N)` then the least value of `4k +n -:2` is:

If underset( x rarr 0 )( "lim") ( e^(-nx) + e^(nx) -cos""( nx)/( 2) - kx^(2))/( ( sin x - tan x )) exists and finite , then possible values of 'n' and 'k' is :

lim_ (x rarr0) (tan mx) / (tan nx)

lim_(x rarr0)(1-cos mx)/(1-cos nx), n!=0

lim_(x rarr0)(1-cos mx)/(1-cos nx),n!=0

lim_ (x rarr0) [(1-cos mx) / (1-cos nx)] = (m ^ (2)) / (n ^ (2))

lim_ (x rarr0) (1- (1-sin x) (1-sin2x) ^ (2) ... (1-sin nx) ^ (n)) / (x) = 55

lim_ (x rarr0) (sin (mx ^ (@))) / (sin (nx ^ (@)))

lim_ (x rarr1) (nx ^ (n + 1) -nx ^ (n) +1) / ((e ^ (x) -e ^ (2)) sin pi x)