The function is defined for all real numbers by

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

The function

h

is defined for all real numbers by

h(x)=x^{2/3}

. Analyze the existence of an instantaneous rate of change at

x=0

by answering the questions that follow. (a) Write an explicit expression for

\displaystyle\frac{h(x)-h(0)}{x-0}

and simplify your result. (b) Evaluate

\displaystyle\lim_{x\to0^-}\frac{h(x)-h(0)}{x}

, carefully justifying each step of your limit process. (c) Evaluate

\displaystyle\lim_{x\to0^+}\frac{h(x)-h(0)}{x}

, carefully justifying each step of your limit process. (d) Combine your findings to reach a conclusion about whether

h

possesses a finite instantaneous rate of change at

x=0

. Describe the geometric significance of your result.