calculus - Evaluate $lim_{xto infty} cos (sqrt {x+1})-cos (sqrt x)$

Evaluate

$$\lim_{x\to \infty} \cos (\sqrt {x+1})-\cos (\sqrt x)$$

For this question I have tried using the squeeze theorem and some trigonometric manipulations such as using

$$2\sin A\sin B=\cos (A-B) -\cos (A+B)$$ . But they were of no use. I also tried using the substitutions like $$x=1/y$$ Hence as $x\to \infty$ , $y\to 0$ but still no success.

I could've also tried using L'Hospital rule but I am not able to convert the function in appropriate indeterminate form. Any help would be greatly appreciated