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

limx→0cosx−1x2.cosx−exxn−2 use 1. Hospital rule = −12limx→0cosx−exxn−2=-1/2limx→0      -sinx-ex(n-2)xn-3 is finite ⇒n-3=0⇒n=3