Question

In Problems 43–52, find the limit as x approaches c of the average rate of change of each function from c to x.

$c=-2;f\left(x\right)=4-3x.$

Short answer

The answer is -3.

Step-by-step solution

Step. 1 Given Information

$$\begin{gathered} c=-2; \\ f\left(x\right)=4-3x. \end{gathered}$$

Step. 2 Average rate of change of f. 

The average rate of change of f from -2 to x is

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(x\right)-f(-2)}{x-(-2)}=\frac{(4-3x)-(4-3(-2\left)\right)}{x+2} \\ =\frac{(4-3x)-(4+6)}{x+2}=\frac{-3x-6}{x+2}=\frac{-3(x+2)}{(x+2)}=-3. \end{gathered}$$

Step. 3 Finding limit 

The limit of the average rate of change is

$\underset{x\to -2}{\lim }\frac{∆y}{∆x}=\underset{x\to -2}{\lim }(-3)=-3.$