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

3

Let $l=\underset{x→0}{\lim}\frac{(\cos x-1)(\cos x-e^x)}{x^n}=\underset{x→0}{\lim}\frac{-2\sin^2\frac{x}{2}(\cos x-e^x)}{x^n}$

For n = 1, n = 2, $l$ = 0

For n = 3,

$l=\underset{x→0}{\lim}-2\left(\frac{\sin(x/2)}{x/2}\right)^2.\frac{\cos x-e^x}{x^2.x}=-\frac{1}{2}\underset{x→0}{\lim}\frac{\cos x-e^x}{x}$

$=\frac{1}{2}\underset{x→0}{\lim}(-\sin x-e^x)=-\frac{1}{2}(-e^0)=\frac{1}{2}$

∴ $l$ is finite and non-zero. Hence n = 3.