The integer n for which $\underset{x\to 0}{lim} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}$ is a finite non-zero number, is

We have

$\begin{array}{l}\underset{x\to 0}{lim} \frac{(\cos x-1)\left(\cos x-e^x\right)}{x^n}\\ =\underset{x\to 0}{lim} \left(\frac{1-\cos x}{x^2}\right)\left(\frac{e^x-\cos x}{x^{n-2}}\right)\\ =\underset{x\to 0}{lim} \left(\frac{1-\cos x}{x^2}\right)\left(\frac{e^x-1+1-\cos x}{x}\right)\frac{1}{x^{n-3}}\\ =\underset{x\to 0}{lim} \left(\frac{1-\cos x}{x^2}\right)\left\{\frac{e^x-1}{x}+\frac{1-\cos x}{x}\right\}\frac{1}{x^{n-3}}\end{array}$