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Stuck here, help me understand: - Limit , continuity and differentiability - JEE Main-4

The set of points where  f\left ( x \right )= \frac{x}{1+\left | x \right |} is differentiable, is

  • Option 1)

    \left ( -\infty ,0 \right )\cup \left ( 0,\infty \right )

  • Option 2)

    \left ( -\infty ,-1 \right )\cup \left ( -1,\infty \right )

  • Option 3)

    \left ( -\infty ,-\infty \right )

  • Option 4)

    \left ( 0 ,\infty \right )

 
Answers (1)
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As we learnt in

Condition for differentiable -

A function  f(x) is said to be differentiable at  x=x_{\circ }  if   Rf'(x_{\circ })\:\:and\:\:Lf'(x_{\circ })   both exist and are equal otherwise non differentiable

-

 

 

 

f(x) =\frac{x}{1+\left | 1+x \right |}

f(x)=\left\{\begin{matrix} \frac{x}{1+x}, x>0 \\ 0,x=0 \\ \frac{x}{1-x}, x<0 \end{matrix}\right.

f'(x)=\left\{\begin{matrix} \frac{1}{({1+x})^2}, x>0 \\ \frac{1}{({1-x})^2}, x<0 \end{matrix}\right.

So f'(0^+)= f'(0)

\therefore f(x) is differentiable for each x \epsilon R

                     


Option 1)

\left ( -\infty ,0 \right )\cup \left ( 0,\infty \right )

Incorrect

Option 2)

\left ( -\infty ,-1 \right )\cup \left ( -1,\infty \right )

Incorrect

Option 3)

\left ( -\infty ,-\infty \right )

Correct

Option 4)

\left ( 0 ,\infty \right )

Incorrect

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