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Question
If \[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]is continuous at x = 0, write the value of k.
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Solution
Given,
\[f\left( x \right) = \begin{cases}\frac{\sin^{- 1} x}{x}, & x \neq 0 \\ k , & x = 0\end{cases}\]
If \[f\left( x \right)\] is continuous at \[x = 0\] , then
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[\Rightarrow \lim_{x \to 0} \left( \frac{\sin^{- 1} x}{x} \right) = k\]
\[ \Rightarrow k = 1 \left[ \because \lim_{x \to 0} \left( \frac{\sin^{- 1} x}{x} \right) = 1 \right]\]
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