Let be a function that is differentiable at with and .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let hh be a function that is differentiable at x=0x=0 with h(0)=5h(0)=5 and h′(0)=−2h'(0)=-2. (a) Evaluate lim⁡x→0h(x)−5x\displaystyle \lim_{x\to 0}\frac{h(x)-5}{x}. Justify each step of your reasoning. (b) Determine lim⁡x→0h(x)\displaystyle \lim_{x\to 0} h(x) and explain how your answer is consistent with the result from part (a). (c) Evaluate lim⁡x→0h(x)−h(−x)x\displaystyle \lim_{x\to 0}\frac{h(x)-h(-x)}{x}, showing all necessary reasoning and algebraic manipulation.

Related theory

Browse the problem library