Question

In Problems 43-52, find the limit as $x$approaches $c$of the average rate of change of each function from $c$to $x$.

$f\left(x\right)=\frac{1}{x^2}$

Short answer

The limit is$-2$

Step-by-step solution

Given information

Given function$f\left(x\right)=\frac{1}{x^2}$

Calculate the average rate of change using the formula

Calculating, we get

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