Question

Evaluate: $\underset{x\to 1}{\lim }{\cos }^{-1}\left(\frac{1-\sqrt{x}}{1-x}\right)$

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{6}$
• C. $\frac{\mathrm{\pi }}{2}$
• D. $\frac{\mathrm{\pi }}{4}$

Answer 1

The correct option is A

$\frac{\mathrm{\pi }}{3}$

Explanation for correct option:

Find the value of $\underset{x\to 1}{\lim }{\cos }^{-1}\left(\frac{1-\sqrt{x}}{1-x}\right)$

Consider the given Equation as:

$$\begin{gathered} I=\underset{x\to 1}{\lim }{\cos }^{-1}\left(\frac{1-\sqrt{x}}{1-x}\right) \\ \Rightarrow I=\underset{x\to 1}{\lim }{\cos }^{-1}\left(\frac{1-\sqrt{x}}{{\left(1\right)}^2-{\left(\sqrt{x}\right)}^2}\right) \\ \Rightarrow I=\underset{x\to 1}{\lim }{\cos }^{-1}\left(\frac{1-\sqrt{x}}{\left(1-\sqrt{x}\right)\left(1+\sqrt{x}\right)}\right) \\ \Rightarrow I=\underset{x\to 1}{\lim }{\cos }^{-1}\left(\frac{1}{\left(1+\sqrt{x}\right)}\right) \\ \Rightarrow I={\cos }^{-1}\left(\frac{1}{\left(1+\sqrt{1}\right)}\right) \\ \Rightarrow I={\cos }^{-1}\left(\frac{1}{\left(1+1\right)}\right) \\ \Rightarrow I={\cos }^{-1}\left(\frac{1}{2}\right) \\ \Rightarrow I=\frac{\mathrm{\pi }}{3}\left[\text{Where},{\cos }^{-1}\left(\frac{1}{2}\right)=\frac{\mathrm{\pi }}{3}\right] \end{gathered}$$

Hence, the correct answer is Option A.