The integer n for which  Ltx→0cosx−1cosx−exxn is a finite nonzero number is

cosx−1cosx−ex=1−x22!+x44!−⋅⋅⋅−11−x22!+x44!−⋅⋅⋅−1+x+x22!+x33!⋅⋅⋅=x2−12+x24!⋅⋅⋅−x−x2−x33!⋅⋅⋅=x3−12+x24!⋅⋅⋅−1−x−x23!⋅⋅⋅(cos⁡x−1)cos⁡x−exx3=12+ terms containing x and higher powers of xLtx→0cosx−1cosx−exx3=12 ⇒n=3.