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: $f(x)=3x^3-2x^2+4$
Thus, we find the average rate of change as:
$\lim\limits_{x\to 0}\dfrac{f(x)-f(0)}{x-0}=\lim\limits_{x\to -1}\dfrac{(3x^3-2x^2+4)-[3(0)^3-2(0)^2+4]}{x} \\=\lim\limits_{x\to 0}\dfrac{x(3x^2-2x)}{x}$
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.
$\lim\limits_{x\to 0} (3x^2-2x) \\=(3)(0)^2-2(0) \\=0 $
