f(x) = (x) / (1 + |x|)
Show from the definition that f: R -> R is differentiable at 0 and find the value of f'(0)
I have been trying to do it from first principles but keep getting no where.
Any help would be appreciated
Note that
$\displaystyle f(0)=0$
Using the limit definition. Taking the limit from the right gives
$\displaystyle f'(0)=\lim_{h \to 0^+}\frac{\frac{h}{1+h}-0}{h}=\lim_{h \to 0^+}\frac{1}{1+h}=1$
Taking the limit from the left
$\displaystyle f'(0)=\lim_{h \to 0^-}\frac{\frac{h}{1-h}-0}{h}=\lim_{h \to 0^-}\frac{1}{1-h}=1$