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  • The set of all points, where the function  is differenti

The set of all points, where the function \mathrm{ f(x)=\frac{x}{1+|x|} } is differentiable is :

Option: 1

\mathrm{ (-\infty, \infty) }


Option: 2

\mathrm{[0, \infty) }


Option: 3

\mathrm{ (-\infty, 0) \cup(0, \infty) }


Option: 4

\mathrm{ (0, \infty) }


Answers (1)

best_answer

\mathrm{ f(x)=\frac{x}{1+|x|} }  is differentiable everywnere except probably at x=0

For, x=0, f(0)=0
\mathrm{\begin{aligned} & L f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(0-h)-f(0)}{-h}=\lim _{h \rightarrow 0} \frac{\frac{-h}{1+h}-0}{-h} \\ & R f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}=\lim _{h \rightarrow 0} \frac{\frac{h}{1+h}-0}{h} \approx 1 \\ & L f^{\prime}(0)=R f^{\prime}(0) \end{aligned} }
\mathrm{\Rightarrow f is differentiable at x=0.}
\mathrm{Hence, f is differentiable in (-\infty, \infty) }

??????

Posted by

vishal kumar

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