Let $f(x)=x\cos(x)$, Which of following is true? 
  
*
  
*Let $f(x) = x\cos x$ for $x\in\mathbb R$. Then 
  $\quad$(A) There is a sequence $x_n\to -\infty$ such that $f(x_n)\to 0$.
  $\quad$(B) There is a sequence $x_n \to \infty$ such that $f(x_n)\to\infty.$
  $\quad$(C) There is a sequence $x_n \to\infty$ such that $f(x_n)\to-\infty$.
  $\quad$(D) $f$ is a uniformly continuous function. 
  

Source.
If I take sequence $-n$, first option is true. What about other options?
Thanks.

A) is true , if you pick $x_n = \pi/2 - n\pi$ you get $f(X)=0$ for every $x$ real. B) is true, if you pick $x_n = 2n\pi$ $f$ goes to $+\infty$ since $cos=1$. $C)$ is true, if you pick $x_n = (2n + 1)\pi$ $f$ goes to $-\infty$ since $cos = -1$. D) is false cause $f'(X) = cosX - XsinX \simeq -X$ for the right choise of $X$ becoming bigger and bigger.
