How to compute $\lim\limits_{x \to 1^+} \left(\frac{\tan \sqrt {x-1}}{\sqrt{x^2-1}}\right)$? I have a problem with this limit, I don't know what method to use. I have no idea how to compute it.
Can you explain the method and the steps used? Thanks
$$\lim\limits_{x \to 1^+} \left(\frac{\tan \sqrt {x-1}}{\sqrt{x^2-1}}\right)$$

Hint:
\begin{align}
\lim_{x\to  1^+}\frac{\tan \sqrt{x-1}}{\sqrt{x^2-1}}&=\left(\lim_{x\to1^+}\frac{\sin\sqrt{x-1}}{\sqrt{x-1}}\right)\left(\lim_{x\to1^+}\frac{1}{(\cos\sqrt{x-1})\sqrt{x+1}}\right)\\
\end{align}
