Limit evaluation for oscillating function I want to evaluate the following oscillating limit x tends to infinity $$\sin(\sqrt{x+1})-\sin(\sqrt{x})$$
I tried evaluating this limit using trigonometric transformations but didn't arrive at the answer

Let $x=\cot^22y$ as $x\to\infty,y\to0^+$
$$\lim_{x\to\infty}(\sqrt{x+1}-\sqrt x)=\lim_{y\to0^+}(\csc2y-\cot2y)=\lim_{y\to0^+}\dfrac{2\sin^2y}{2\sin y\cos y}$$
Cancel out $\sin y$ safely as $y\to0,y\ne0,\sin y\to0,\sin y\ne0$
