Let . (i) Write the algebraic expressions for the difference quotient separately for…

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let f(x)=∣x∣f(x)=|x|. (i) Write the algebraic expressions for the difference quotient f(0+h)−f(0)h\dfrac{f(0+h)-f(0)}{h} separately for (a) h>0h>0 and (b) h<0h<0. (ii) Evaluate lim⁡h→0+f(0+h)−f(0)h\displaystyle \lim_{h\to 0^{+}} \dfrac{f(0+h)-f(0)}{h} and lim⁡h→0−f(0+h)−f(0)h\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)f'(0) exists. Give a brief reason for your answer.

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