k का वह मान क्या है जो  \(f\left( x \right)\; = \;\left\{ {\begin{array}{*{20}{c}} {\sin x\;,x \ne 0}\\ {k\;,x\; = \;0} \end{array}} \right.\) को x = 0 पर संतत बनाता है?

धारणा:

f(x) x = 0 पर निरंतर है

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

गणना:

f(x) x = 0 पर निरंतर है

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

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

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

\(\Rightarrow k = 0\)

\(\therefore \;k = 0\)