Double Limit

Calculus Level 3

$\large \lim_{m \to \infty} \lim_{n \to \infty} {(1 + \cos^{2m}(n! \pi x))}$

Find the limit above, if $x$ is irrational.

The answer is 1.

1 solution

Chew-Seong Cheong
Nov 14, 2016

$\begin{aligned} \lim_{m \to \infty} \lim_{n \to \infty} \left(1+ \cos^{2m}(n!\pi x) \right) & = \lim_{m \to \infty} \left(1+ |\cos (n!\pi x)|^{2m} \right) & \small {\color{#3D99F6}\text{For irrational }x, \ -1 < \cos (n!\pi x) < 1 } \\ & = 1 + 0 = \boxed{1} \end{aligned}$

Double Limit

The answer is 1.

1 solution

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