Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 388: 118

$$1$$

Since \begin{align*} \lim _{x \rightarrow 1} \frac{\sqrt{1-x^{2}}}{\cos ^{-1} x}&=\frac{0}{0} \end{align*} Then by using L’Hôpital’s Rule \begin{align*} \lim _{x \rightarrow 1} \frac{\sqrt{1-x^{2}}}{\cos ^{-1} x}\\ &= \lim _{x\to \:1}\left(\frac{-\frac{x}{\sqrt{1-x^2}}}{-\frac{1}{\sqrt{1-x^2}}}\right)\\ &= \lim _{x\to \:1}\left(x\right)\\ &=1 \end{align*}