Question

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

$$\begin{gathered} c=1; \\ f\left(x\right)=\frac{1}{x} \end{gathered}$$

Short answer

The limit is$-1$

Step-by-step solution

Given information 

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

Calculate the average rate of change using the formula

Calculating, we get

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