Question

find the limit as x approaches c of the average rate of change of each function from c to x.

$c=0;f\left(x\right)=4x^3-5x+8$

Short answer

The limit is$-5$

Step-by-step solution

Step 1. The given information is

$x=0;f\left(x\right)=4x^3-5x+8$

Step 2. An average rate change of function f  from point c to x

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(x\right)-f\left(c\right)}{x-c} \\ \frac{∆y}{∆x}=\frac{4x^3-5x+8-(4c^3-5c+8)}{x-0} \\ \frac{∆y}{∆x}=\frac{4x^3-5x+8-8)}{x} \\ \frac{∆y}{∆x}=\frac{4x^3-5x}{x} \\ \frac{∆y}{∆x}=\frac{x(4x^2-5)}{x} \\ \frac{∆y}{∆x}=4x^2-5 \end{gathered}$$

Step 3. Solving the limit we get,

$$\begin{gathered} \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c} \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to c}{\lim }(4x^2-5) \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=4{\left(0\right)}^2-5 \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=-5 \end{gathered}$$