[tex] \displaystyle \sin(\sqrt{x+1})- \sin(\sqrt{x})=2 \sin \frac{\sqrt{x+1}-\sqrt{x}}{2} \cos \frac{\sqrt{x+1}+ \sqrt{x}}{2}.\\ \\ Observam~ca~\frac{\sqrt{x+1}-\sqrt{x}}{2}= \frac{1}{2(\sqrt{x+1}+ \sqrt{x})}. \\ \\ Deci \lim_{x \to \infty} \frac{\sqrt{x+1}-\sqrt{x}}{2} = 0.\\ \\ Atunci~\lim_{x \to \infty} \sin \frac{\sqrt{x+1}-\sqrt{x}}{2}=0.\\ \\ Intrucat~\cos \frac{\sqrt{x+1}+\sqrt{x}}{2}~variaza~in~intervalul~[-1,1],~rezulta\\ \\ ca~limita~initiala~este~0. [/tex]
