Proving that a limit equal 0 as $x \rightarrow \infty$ by epsilon delta definition.

Question

How can I proof by epsilon delta definition that the limit of this function is 0 as $x \rightarrow \infty$ $$\cos \frac {1}{2(\sqrt{x +1} - \sqrt {x})} \sin \frac {1}{2(\sqrt{x +1} + \sqrt {x})} $$

My attempt : I will start by $$|\cos \frac {1}{2(\sqrt{x +1} - \sqrt {x})} \sin \frac {1}{2(\sqrt{x +1} + \sqrt {x})}| < \epsilon $$

But then what?

Answer 1

Hint: You can use the squeeze theorem since we have $$0\leq\left|\cos\left(\frac{1}{2(\sqrt{x+1}-\sqrt{x})}\right)\sin\left(\frac{1}{2(\sqrt{x+1}+\sqrt{x})}\right)\right|\leq\left|\sin\left(\frac{1}{2(\sqrt{x+1}+\sqrt{x})}\right)\right|$$where I've used the property that $|\cos \theta|\leq1$ for all $\theta\in\Bbb R$. Now you just need to prove that the right-hand side converges to $0$ using the $\varepsilon$-$\delta$ definition.