x->0
cos(xe^x)
=1- (1/2)(xe^x)^2 + (1/24)(xe^x)^4 +o(x^4)
=1- (1/2)[x + x^2 + (1/2)x^3+o(x^3) ]^2 + (1/24)[x^4 +o(x^4)] +o(x^4)
=1- (1/2)[x^2 + x^4 +2x^3 + x^4 +o(x^4) ] + (1/24)[x^4 +o(x^4)] +o(x^4)
=1- (1/2)[x^2 + 2x^3+2x^4+o(x^4) ] + (1/24)[x^4 +o(x^4)] +o(x^4)
=1 -(1/2)x^2 - x^3 - (23/24)x^4 +o(x^4)
e^(2x)= 1+ 2x + 2x^2 +o(x^2)
-(x^2/2).e^(2x) = -(1/2)x^2- x^3 - x^4 +o(x^4)
e^[-(x^2/2).e^(2x)]
= e^[-(1/2)x^2- x^3 - x^4 +o(x^4)]
=1 +[-(1/2)x^2- x^3 - x^4 +o(x^4)] + (1/2){ [-(1/2)x^2- x^3 - x^4 +o(x^4)]}^2
+o(x^4)
=1 +[-(1/2)x^2- x^3 - x^4 +o(x^4)] + (1/2)[(1/4)x^4+o(x^4)]+o(x^4)
=1 -(1/2)x^2 -x^3 - (7/8)x^4 +o(x^4)
cos(xe^x) -e^[-(x^2/2).e^(2x)]
=[1 -(1/2)x^2 - x^3 - (23/24)x^4 +o(x^4) ]-[1 -(1/2)x^2 -x^3 - (7/8)x^4 +o(x^4)]
=(-23/24 +7/8)x^4 +o(x^4)
=-(1/12)x^4 +o(x^4)
lim(x->0) { cos(xe^x)- e^[-(x^2/2).e^(2x) ] }/ x^4
=lim(x->0) -(1/12)x^4/ x^4
=-1/12
