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Please solve RD Sharma class 12 chapter 8 Continuity exercise Very short answer question 9 maths textbook solution

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Answer: 1

Hint: \lim _{x \rightarrow 0} f(x)=f(0)

Given: f(x)=\left\{\begin{array}{l} \frac{\sin ^{-1} x}{x} \\ k, x=0 \end{array}, x \neq 0\right. is continuous at x = 0


                    Asf(x) is continuous at x = 0

                    \begin{aligned} &\lim _{x \rightarrow 0} f(x)=f(0) \\ &\lim _{x \rightarrow 0} \frac{\sin ^{-1} x}{x}=k \end{aligned}

                    Applying L’Hospital rule as the function is in 0/0 form,

                    \begin{aligned} &\lim _{x \rightarrow 0} \frac{1}{\frac{\sqrt{1-x^{2}}}{1}}=k \end{aligned}                                            \left [ \frac{0}{0}form \right ]

                    \begin{aligned} &\frac{1}{\sqrt{1-0}}=k \\ &\therefore k=1 \end{aligned}

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