Answer
$-2$
Work Step by Step
General formula for average rate of change from $c$ to $x$ can be written as: $\dfrac{f(x)-f(c)}{x-c}$
We have: $f(x)=\dfrac{1}{x^2}$
In order to simplify the above expression, we will use the following rules.
$(a) \lim\limits_{x \to c} \dfrac{a(x)}{b(x)}=\dfrac{\lim\limits_{x \to c} a(x)}{\lim\limits_{x \to c} b(x)} \\ (b) \lim\limits_{x \to c} p(x)=p(c)$ ;
where $a$ as a constant.
$\lim\limits_{x\to 1}\dfrac{f(x)-f(1)}{x-1}=\lim\limits_{x\to 1}\dfrac{\dfrac{1}{x^2}-1}{x-1} \\=\lim\limits_{x\to 1}\dfrac{\dfrac{1-x^2}{x^2}}{x-1} \\=\dfrac{\lim\limits_{x\to 1}-(x-1)(x+1)}{\lim\limits_{x\to 1} x^2(x-1)} \\=\dfrac{\lim\limits_{x\to 1} -(x+1)} {\lim\limits_{x\to 1} x^2} \\=-\dfrac{1+1}{1} \\=-2 $
