Question

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

$c=-1;f\left(x\right)=x^2+2x$

Short answer

Limit is$0$

Step-by-step solution

Given information is

$c=-1;f\left(x\right)=x^2+2x$

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

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

Step 3. Solving we get,

$$\begin{gathered} \frac{∆y}{∆x}=\frac{x^2+2x+1}{x+1} \\ \frac{∆y}{∆x}=\frac{{\left(x\right)}^2+2\times 1\times x+1^2}{x+1} \\ \frac{∆y}{∆x}=\frac{{(x+1)}^2}{x+1} \\ \frac{∆y}{∆x}=(x+1) \end{gathered}$$

Step 4. Finding the limit we get,

$$\begin{gathered} \underset{x\to c}{\lim }\frac{∆y}{∆x}=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c} \\ \underset{x\to -1}{\lim }\frac{∆y}{∆x}=\underset{x\to -1}{\lim }(x+1) \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=-1+1 \\ \underset{x\to c}{\lim }\frac{∆y}{∆x}=0 \end{gathered}$$