Question
Download Solution PDFThe 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
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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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