If $x$ is a real number in [0, 1], then the value of $\mathop {\lim }\limits_{m \to \infty } \mathop {\lim }\limits_{n \to \infty } \left[ {1 + {{\cos }^{2m}}\left( {n!\pi {\rm{ }}x} \right)} \right]$ is given by

1 or 2 according as $x$ is rational or irrational

JEE MATHEMATICS MCQ: LIMITS

1 Medium

1 of 1

Analyzing +4 | -1

x

is a real number in [0, 1], then the value of

\lim\limits_{m \to \infty } \lim\limits_{n \to \infty } \left[ {1 + {{\cos }^{2m}}\left( {n!\pi \mathrm{ }x} \right)} \right]

is given by

View answer and explanation

Correct option: A

2 or 1 according as

x

is rational or irrational

Option A (2 or 1 according as

x

is rational or irrational) is correct. When x is rational, the limit is 2; when x is irrational, the limit is 1.