Question

The value of $\underset{m\to \infty }{lim}\underset{n\to \infty }{lim}[2+{\cos }^{m!}(n!\pi x\left)\right],x\in R$, where $n,m\in Z$ is/are

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (C), (D)

The correct options are
C $2$
D $3$

Case 1:
When $n!x$ is a integer
$\Rightarrow {\cos }^{m!}(n!\pi x)=(\pm 1{)}^{m!}=1$
($\because m!$ is even)
$\Rightarrow \underset{m\to \infty }{lim}\underset{n\to \infty }{lim}[2+{\cos }^{m!}(n!\pi x\left)\right]=2+1=3$

Case 2:
When $n!x$ is a non integer
$\Rightarrow -1<\cos (n!\pi x)<1$
$\Rightarrow \underset{m\to \infty }{lim}\underset{n\to \infty }{lim}[2+{\cos }^{m!}(n!\pi x\left)\right]=2+0=2$