Mathematics

Problem

Let

f(x)=|x|

. a. (i) Compute the right–hand limit

L_{+}=\displaystyle\lim_{h\to 0^{+}}\frac{f(0+h)-f(0)}{h}

and the left–hand limit

L_{-}=\displaystyle\lim_{h\to 0^{-}}\frac{f(0+h)-f(0)}{h}

. Keep your answers in simplest numerical form. b. (ii) On the basis of the values of

L_{+}

and

L_{-}

obtained in part (i), decide whether the derivative of

f

exists at

x=0

. Give a brief reason for your conclusion.