The integer n for which limx→0(cosx−1)(cosx−ex)xn is a finite nonzero number is

Given that, limx→0(cosx−1)(cosx−ex)xn= finite non zero number =limx→0(cosx−1)(1+cosx)(ex−cosx)xn(1+cosx) =limx→0sin2xx2.ex−cosxxn−2.11+cosx  =12.12limx→0[1+x1!+x22!+x33!+....∞]−[1−x22!+x44!+x66!+....∞]xn−2 =12limx→0(1+x+x23!+2x34!+.....∞)xn−3 for this limit to be finite n-3 = 0 ⇒n = 3