$\lim\limits_{m \rightarrow \infty} \lim\limits_{n \rightarrow \infty}\left(1+\cos ^{2 m} n ! x \pi\right)$ is equal to (x is rational):

2

We know that $|\cos \theta| \leq 1$ for all $\theta$. Also $|\cos n ! \pi x|<1$, if $x$ is irrational.

Hence $\lim\limits_{m \rightarrow 0} \lim\limits_{n \rightarrow 0}\left(1+\cos ^{2 m} n ! \pi x\right)=1$ and if x is rational (i.e. $x=\frac{p}{q} p, q ~€~ I$) . $n ! x \pi$ is an integral multiple of $\pi$. Hence $\cos n ! \pi x=1$ or -1 and $\cos ^{2 m} n ! x \pi=1$.

Hence $\lim\limits_{m \rightarrow 0} \lim\limits_{n \rightarrow 0}\left(1+\cos ^{2 m} n ! \pi x\right)=2$

Hence (1) is the correct answer.