I need help with some limit I got stuck with. $$\lim_{x\to +\infty} \frac{\cos\left(\frac{\sqrt{x+1}+\sqrt{x}}{2}\right)}{\sin\left(\frac{\sqrt{x+1}-\sqrt{x}}{2}\right)} - 4\sqrt{x+1}\cos\left(\sqrt{x+1}\right)-\sin\left(\sqrt{x+1}\right)$$
When I calculate this for large numbers, it seem like it tends toward zero. As an attempt to prove it, I squared the equation in WolframAlpha and after some simplifying: $$\left(\frac{\cos\left(\frac{\sqrt{x+1}+\sqrt{x}}{2}\right)}{\sin\left(\frac{\sqrt{x+1}-\sqrt{x}}{2}\right)} - 4\sqrt{x+1}\cos\left(\sqrt{x+1}\right)-\sin\left(\sqrt{x+1}\right)\right)^2 = \frac{1+\cos\left(2\sqrt{x+1}\right)}{2}\left(4\sqrt{x+1}-\cot\left(\frac{\sqrt{x+1} - \sqrt{x}}{2}\right)\right)^2$$
noticing that: $$0 \leq \frac{1+\cos\left(2\sqrt{x+1}\right)}{2} \leq 1$$
so that means the above limit is related to this new limit: $$\lim_{x\to +\infty} 4\sqrt{x+1}-\cot\left(\frac{\sqrt{x+1} - \sqrt{x}}{2}\right)$$ which also seem to tend toward zero but don't know how to prove it, any help?