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Determine the Value of the Constant K So that the Function F ( X ) = { Sin 2 X 5 X , I F X ≠ 0 K , I F X = 0 is Continuous at X = 0 .

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Question

Determine the value of the constant k so that the function

\[f\left( x \right) = \begin{cases}\frac{\sin 2x}{5x}, if & x \neq 0 \\ k , if & x = 0\end{cases}\text{is continuous at x} = 0 .\]

 

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Solution

\[f\left( x \right) = \binom{\frac{\sin2x}{5x}, if x \neq 0}{k, if x = 0}\]

If  

\[f\left( x \right)\]  is continuous at x = 0, then
\[\lim_{x \to 0} f\left( x \right) = f\left( 0 \right)\]
\[\lim_{x \to 0} \frac{\sin2x}{5x} = k\]
\[\lim_{x \to 0} \frac{2\sin2x}{5 \times 2x} = k\]
\[\frac{2}{5} \lim_{x \to 0} \frac{\sin2x}{2x} = k\]
\[\frac{2}{5} \times 1 = k\]
\[k = \frac{2}{5}\]
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Chapter 8: Continuity - Exercise 9.1 [Page 19]

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R.D. Sharma Mathematics Volume 1 and 2 [English] Class 12
Chapter 8 Continuity
Exercise 9.1 | Q 22 | Page 19

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