Mathematics

Problem

Let

f(x)=|x|

. (i) Write the algebraic expressions for the difference quotient

\dfrac{f(0+h)-f(0)}{h}

separately for (a)

h>0

and (b)

h<0

. (ii) Evaluate

\displaystyle \lim_{h\to 0^{+}} \dfrac{f(0+h)-f(0)}{h}

and

\displaystyle \lim_{h\to 0^{-}} \dfrac{f(0+h)-f(0)}{h}

. (iii) On the basis of the results obtained in (ii), state whether

f'(0)

exists. Give a brief reason for your answer.