Can You Simplify Limits Using Big O Notation?

[catapax]

lim (e^x-sin(x)-cos(x))/(e^(x^2)-e^(x^3)), x->0
= lim (1+x+o(x)-x+o(x)+1/2(x^2)-1+o(x^2))/((x^2)+1+o(x^2)-1-(x^3)+o(x^3)), x->0
= lim 1/2(x^2)/(x^2+o(x^2)), x->0 = 1/2
is it correct?

 

[mfb]

You cannot simplify o(x)+1/2 x^2 to 1/2 x^2, as o(x) could (and does in your example!) include terms of x^2. You have to expand it more.