Is $\sup_{x\in \mathbb R} \sqrt{|x|} \sin x = + \infty$?

Question

The graph of $\sqrt{|x|} \sin x$ seems like staying around the $x$-axis, but $\lim_{x\to \infty} \sqrt{|x|} = +\infty$ and $\sin x$ is bounded.

Answer 1

Let $x = \frac{\pi}2 +2n\pi$,

Then, $$\sqrt{|x|} \sin x=\sqrt{\frac{\pi}2+2n\pi}$$

Now, let $n\to \infty$ to get your result.