$\displaystyle

\lim _{x->0} \frac{cos(xe^x)-cos(xe^{-x})}{x^3}\\

$

$\displaystyle

e^x=1+x+O(x^2)\\

$

$\displaystyle

e^{-x}=1-x+O(x^2)\\

$

$\displaystyle

xe^x=x+O(x^2)\\

$

$\displaystyle

cos(x)=1-\frac{1}{2!}x^2+O(x^3)\\

$

$\displaystyle

\lim_{x->0} \frac{1-\frac{1}{2!}(x+O(x^2))^2+O(x^3)-1+\frac{1}{2!}(x+O(x^2))^2+O(x^3)}{x^3}

$

but i dont know how to deal with the remainders

there squaring of them etc..

??