Chapter 2: Limits and Continuity - Section 2.2 - Limit of a Function and Limit Laws - Exercises 2.2 - Page 57: 49

$\sqrt{-\pi+4}$

Note that, $\lim\limits_{x \to a}\cos x=\cos a$ for all real numbers $a$. Using this along with our limit laws, we get $\lim\limits_{x \to -\pi}[(\sqrt{x+4})\cos (x+\pi)]=(\sqrt{-\pi+4})\cos 0=(\sqrt{-\pi+4})(1)=\sqrt{-\pi+4}.$