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

3

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

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

The limit is non-zero

if n - 2 = 1 ⇒ n = 3.

Hence (C) is the correct answer.

Alternate:

$\underset{x→0}{\lim}\frac{(\cos x-1)(\cos x-e^x)}{x^n}=\frac{(-\frac{x^2}{2!}+\frac{x^4}{4!}+.....)(-x-x^2-...)}{x^n}$ is non zero for n = 3.