First slide

Introduction to limits

Question

If x is a real number in $[0, 1]$ then the value of
$lim_{n \to \infty} lim_{m \to \infty} [1 + \cos^{2 m} (n! π x)]$ is given by

Moderate

A

2 or 1 according as x is rational or irrational
B

1 or 2 according as x is rational or irrational
C

1 for all $x$
D

2 or 1 for all x

Solution

If $x \in Q$ then $n! π x$ will be an integral multiple of $π$
for large values of $n$ . Therefore, $\cos (n! π x)$ will be either 1 or
-1 and so $\cos^{2 m} (n! π x) = 1$
$lim_{m \to \infty} lim_{n \to \infty} [1 + \cos^{2 m} (n! π x)] = 1 ∣ + 1 = 2$
If $x \to Q, n! π x$ will not be an integral multiple of $π$ and so
$\cos (n! π x)$ will lie between -1 and 1
thus, $lim_{m \to \infty} \cos^{2 m} (n! π x) = 0$
$\Rightarrow lim_{m \to \infty} lim_{n \to \infty} [1 + \cos^{2 m} (n! π x)] = 1 + 0 = 1 .$
Thus, option (a) is correct.

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