\lim_{x\to \infty }  \text{Sin}\left[\sqrt{1+x}\right]-\text{Sin}\left[\sqrt{x}\right]\text{==}\lim_{x\to \infty } 2 \text{Cos}\left[\frac{1}{2} \left(\sqrt{x}+\sqrt{1+x}\right)\right] \text{Sin}\left[\frac{1}{2} \left(\sqrt{1+x}-\sqrt{x}\right)\right]