Question

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

$c=3;f\left(x\right)=x^3$

Short answer

Limit is$9$

Step-by-step solution

Step 1. Given information is

$c=3;f\left(x\right)=x^3$

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

$$\begin{gathered} \frac{∆y}{∆x}=\frac{f\left(x\right)-f\left(c\right)}{x-c} \\ \frac{∆y}{∆x}=\frac{x^3-c^3}{x-c} \\ \frac{∆y}{∆x}=\frac{(x-3)(x^2-3x+3^2)}{x-3} \\ \frac{∆y}{∆x}=\frac{(x^2-3x+3^2)}{1} \\ =x^2-3x+9 \end{gathered}$$

Step 3. Now, finding the limit we get,

$$\begin{gathered} \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c} \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to 3}{\lim }(x^2-3x+9) \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}={\left(3\right)}^2-3\left(3\right)+9 \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=9-9+9 \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=9 \end{gathered}$$