Chapter 3 - Section 3.5 - Derivatives of Trigonometric Functions - Exercises - Page 152: 48

$$\lim_{x\to-\pi/6}\sqrt{1+\cos(\pi+\csc x)}=\sqrt{2}$$

$$A=\lim_{x\to-\pi/6}\sqrt{1+\cos(\pi\csc x)}=\lim_{x\to-\pi/6}\sqrt{1+\cos\Big(\frac{\pi}{\sin x}\Big)}$$ $$A=\sqrt{1+\cos\Big(\frac{\pi}{\sin(-\pi/6)}\Big)}$$ $$A=\sqrt{1+\cos\Big(\frac{\pi}{-\frac{1}{2}}\Big)}$$ $$A=\sqrt{1+\cos(-2\pi)}$$ $$A=\sqrt{1+1}=\sqrt2$$