Answer
$$0$$
Work Step by Step
The general formula for average rate of change from $a$ to $b$ can be written as: $\dfrac{f(b)-f(a)}{b-a}$
Here, we have: $f(x)=x^2+2x=x(x+2)$
Thus, we find the average rate of change as:
$\lim\limits_{x\to -1}\dfrac{f(x)-f(-1)}{x-(-1)}=\lim\limits_{x\to -1}\dfrac{x(x+2)-[(-1)^2+2(-1)]}{x+1} \\=\lim\limits_{x\to -1}\dfrac{x^2+2x+1}{x+1}$
In order to simplify the above expression, we will use the following rules.
$(a) \lim\limits_{x \to a} \dfrac{p(x)}{q(x)}=\dfrac{\lim\limits_{x \to a} p(x)}{\lim\limits_{x \to a} q(x)} \\ (b) \lim\limits_{x \to a} k(x)=k(a)$ ;
where $a$ is a constant.
$\dfrac{\lim\limits_{x\to -1} (x+1)^2}{\lim\limits_{x\to -1} (x+1)} \\=\lim\limits_{x\to -1} (x+1) \\=-1+1 \\=0$
