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For the function f (x) = {{:(x/(1+e^(1...

For the function `f (x) = {{:(x/(1+e^(1//x))", " x ne 0),(" 0 , "x = 0):}`, the derivative from the right, `f'(0^(+)) `= … and the derivative from the left, ` f'(0^(-))` = … .

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The correct Answer is:
`f'(0^(+)) = 0, f'(0^(-)) = 1`
Given, ` f(x) ={{:(x/(1+e^(1//x))", " x ne 0),(" 0, " x= 0):}`
`:. Rf'(0) = f'(0^(+)) = underset( h to 0) lim (1+e^(1//h))/h = underset(h to 0) lim 1/(1+e^(1//h)) = 0 `
and `Lf'(0) = f'(0^(-)) = underset( h to 0) lim ((-h)/(1+e^(-1//h)))/(-h)= underset( h to 0) lim 1/(1+1/e^(1//h)) = 1/(1+0) = 1 `
` :. f'(0^(+)) = 0 and f'(0^(-)) = 1`
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