Answer
$\lim\limits_{x\to\pi^+}\dfrac{\sqrt{x}}{\csc{x}}=0.$
Work Step by Step
$\lim\limits_{x\to\pi^+}\dfrac{\sqrt{x}}{\csc{x}}\to$
$\lim\limits_{x\to\pi^+}\sqrt{x}=\sqrt{\pi^+}=\sqrt{\pi}$
$\lim\limits_{x\to\pi^+}\csc{x}=\csc{\pi^+}=-\infty.$
By Theorem $1.15:$
$\lim\limits_{x\to\pi^+}\dfrac{\sqrt{x}}{\csc{x}}=-\dfrac{\sqrt{\pi}}{\infty}=0.$
