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The value of k which makes \(f\left( x \right)\; = \;\left\{ {\begin{array}{*{20}{c}} {\sin x\;,x \ne 0}\\ {k\;,x\; = \;0} \end{array}} \right.\) continuous at x = 0, is
1. 2
2. 1
3. -1
4. 0
5. None of these

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Correct Answer - Option 4 : 0

Concept:

f(x) is Continuous at x = 0

\(\Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right)\; = \;\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right)\; = f(0)\)

Calculation:

f(x) is Continuous at x = 0

\(\Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} f\left( x \right)\; = \;\mathop {\lim }\limits_{x \to {0^ - }} f\left( x \right)\; = f(0)\)

\( \Rightarrow \mathop {\lim }\limits_{x \to {0^ + }} \sin x\; = \;\mathop {\lim }\limits_{x \to {0^ - }} \sin x\; = k\)

\(\Rightarrow \mathop {\lim }\limits_{h \to 0} \sin \left( {0 + h} \right)\; = \;\mathop {\lim }\limits_{h \to 0} \sin \left( {0 - h} \right)\; =k\)

\(\Rightarrow k = 0\)

\(\therefore \;k = 0\)

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