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

$\underset{x\to 0}{\lim }\frac{-(1-\cos x)\left(\cos x-e^x\right)}{x^2\cdot x^{n-2}}=\frac{-1}{2}\underset{x\to 0}{\lim } \frac{\cos x-e^x}{x^{n-2}}$$$\begin{gathered} =\frac{-1}{2}\cdot \underset{\mathrm{x}\to 0}{\lim } \frac{-\sin \mathrm{x}-{\mathrm{e}}^{\mathrm{x}}}{(\mathrm{n}-2){\mathrm{x}}^{\mathrm{n}-3}} \\ \therefore \mathrm{n}-3=0\Rightarrow n=3 \end{gathered}$$