Calculate:$\lim_{x \rightarrow (-1)^{+}}\left(\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}} \right)$ How to calculate following with out using L'Hospital rule
$$\lim_{x \rightarrow (-1)^{+}}\left(\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}} \right)$$

\begin{eqnarray*}
\lim_{x\rightarrow\left(-1\right)^{+}}\frac{\sqrt{\pi}-\sqrt{\arccos x}}{\sqrt{x+1}} & = & \lim_{x\rightarrow\left(-1\right)^{+}}\frac{\pi-\arccos x}{2\sqrt{\pi}\sqrt{x+1}}\\
 & = & \lim_{x\rightarrow\left(-1\right)^{+}}\frac{\sin\left(\pi-\arccos x\right)}{2\sqrt{\pi}\sqrt{x+1}}\\
 & = & \lim_{x\rightarrow\left(-1\right)^{+}}\frac{\sin\arccos x}{2\sqrt{\pi}\sqrt{x+1}}\\
 & = & \lim_{x\rightarrow\left(-1\right)^{+}}\frac{\sqrt{1-x^{2}}}{2\sqrt{\pi}\sqrt{x+1}}\\
 & = & \frac{1}{\sqrt{2\pi}}
\end{eqnarray*}
