Mathematics

Problem

Let

h

be a function that is differentiable at

x=0

with

h(0)=5

and

h'(0)=-2

. (a) Evaluate

\displaystyle \lim_{x\to 0}\frac{h(x)-5}{x}

. Justify each step of your reasoning. (b) Determine

\displaystyle \lim_{x\to 0} h(x)

and explain how your answer is consistent with the result from part (a). (c) Evaluate

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

, showing all necessary reasoning and algebraic manipulation.