$\underset{x\to +\infty }{\mathop{\lim }}\,(\cos \sqrt{x+1}-\cos \sqrt{x})=\underset{x\to +\infty }{\mathop{\lim }}\,(-2\sin \frac{\sqrt{x+1}+\sqrt{x}}{2}\sin \frac{\sqrt{x+1}-\sqrt{x}}{2})$ $=-2\underset{x\to +\infty }{\mathop{\lim }}\,\frac{\sqrt{x+1}+\sqrt{x}}{2}\times \underset{x\to +\infty }{\mathop{\lim }}\,\sin \frac{\sqrt{x+1}-\sqrt{x}}{2}$  $\underset{x\to +\infty }{\mathop{\lim }}\,\sin (\frac{\sqrt{x+1}-\sqrt{x}}{2}\times \frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}})=\underset{x\to +\infty }{\mathop{\lim }}\,\sin \frac{x+1-x}{2(\sqrt{x}+\sqrt{x})}=\sin \frac{1}{+\infty }=\sin 0=0$