Answer
$0$
Work Step by Step
Definition of a sequence defined by a function: Let us consider a function $f(x)$ whose limit $\lim\limits_{x \to \infty} f(x)$ exists then the sequence $a_n=f(n)$ will converge to the same limit.
That is, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} f(x)$
Here, in this problem we have $a_n=n [1-\cos (1/n)]$ and $f(x)=x [1-\cos (1/x)]$
Next, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} x [1-\cos (1/x)]$
or, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} \dfrac{1-\cos (1/x)}{1/x}=\lim\limits_{x \to \infty} \dfrac{\sin^2(1/2x)}{1/2x}$
or, $\lim\limits_{n \to \infty} a_n=\lim\limits_{x \to \infty} \dfrac{\sin^2(1/2x)}{(1/2x)^2} \times \dfrac{1}{2x}$
Thus. $\lim\limits_{n \to \infty} a_n=(1)(0)=0$
