First slide

Introduction to limits

Question

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

Moderate

A

2 or 1 according as x is a rational or irrational
B

2 or 1 according as x is irrational or rational
C

1 for all x
D

2 for all x

Solution

$\cos (n! π x) = \pm 1 f o r x i s r a t i o n a l$ $\cos^{2 m} (n! π x) \to 0 a s m \to \infty f o r x i s i r r a t i o n a l$

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