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Explain solution rd sharma class 12 chapter 9 Differentiability exercise Multiple choice question, question 23

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HINTS: check at x=0 LHD & RMD

GIVEN: f(x)=\left\{\begin{array}{cc} \frac{1}{1+e^{\frac{1}{x}}} & x \neq 0 \\ 0 & x=0 \end{array}\right.


At x=0

\lim _{x \rightarrow 0^{-}} f(x)=\lim _{h \rightarrow 0} f(-h)=\lim _{h \rightarrow 0} \frac{1}{1+e^{\frac{1}{-h}}}=1


\begin{aligned} &=\lim _{x \rightarrow \sigma^{+}} f(x)=\lim _{h \rightarrow 0} f(h) \\ &=\lim _{h \rightarrow 0} \frac{1}{1+e^{\frac{1}{-h}}}=\frac{1}{1+e^{\infty}}=0 \end{aligned}

As LHL \neq RHL

f(x) is not continuous, so f(x) is not differentiable

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