The integer ‘n’ for which  $\underset{x\to 0}{Lt}\frac{\cos 2x-1}{x^n}$is a finite non zero number is

$\underset{x\to 0}{Lt}\frac{\cos 2x-1}{x^n}$

by L.H.L

$\underset{x\to 0}{Lt}\frac{-2\sin 2x}{n{x}^{n-1}}$

$\underset{x\to 0}{Lt}\frac{-4\cos 2x}{n(n-1)x^{n-2}}$

$n-2=0\Rightarrow n=2$

for limit to be finite

$n-2=0\Rightarrow n=2$