Answer
$-1$
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: $f(x)=\dfrac{1}{x}$
Thus, we find the average rate of change as:
$\lim\limits_{x\to 1}\dfrac{f(x)-f(1)}{x-1}=\lim\limits_{x\to 1}\dfrac{\dfrac{1}{x}-1}{x-1} \\=\lim\limits_{x\to 1}\dfrac{\dfrac{1-x}{x}}{x-1}$
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)}{\lim\limits_{x\to 1} x(x-1)} \\=\lim\limits_{x\to 1} (-\dfrac{1}{x} ) \\=-1 $
