A) 1
B) ? 1
C) 0
D) Does not exist
Correct Answer: D
Solution :
If \[f(x)=\left\{ \begin{align} & x+2\,,\,\,\,-1<x<3 \\ & \,\,\,\,\,\,5\,\,,\,\,\,\,\,\,\,\,\,\,\,\,x=3 \\ & 8-x\,\,,\,\,\,\,\,\,\,\,\,\,\,x>3 \\ \end{align} \right.\] and \[f(3)=5\] L.H.D =\[\underset{x\to 3-}{\mathop{\lim }}\,\frac{f(x)-f(3)}{x-3}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(3-h)-f(3)}{-h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{(3-h+2)-5}{-h}=\underset{h\to 0}{\mathop{\lim }}\,\frac{-h}{-h}=1\] R.H.D \[=\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,\frac{f(x)-f(3)}{x-3}=\underset{h\to 0}{\mathop{\lim }}\,\frac{f(3+h)-f(3)}{h}\] \[=\underset{h\to 0}{\mathop{\lim }}\,\frac{8-(3+h)-5}{h}=\underset{h\to 0}{\mathop{\lim }}\,\frac{-h}{h}=-1\] L.H.D \[\ne \] R.H.D \[f(x)\] is not differentiable.
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