Minimum $n$

Using the big o notation , we have (as $x \to 0$ ) : $\begin{aligned} \cos x &= 1-\frac{x^2}{2} + O(x^4) \\ \sin x &= x - \frac{x^3}{3!} + O(x^5) \\ e^x + e^{-x} &= 2+x^2+O(x^4) \end{aligned}$

Hence, $(1-\cos x)(\sin x - x)(e^x+e^{-x}-2) = -\frac{x^7}{12} + O(x^9) \quad as \, x \to 0$

So, the required answer is $\boxed{\boxed{7}}$

Use the taylor expansion of each. The lowest powers of x will be taken from each bracket so that limit is finite non zero number. Thus answer is 2+3+2 in order that is 7