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Q. 51

Expert-verified
Found in: Page 883

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# In Problems 43-52, find the limit as $x$ approaches $c$ of the average rate of change of each function from $c$ to $x$.$c=1;\phantom{\rule{0ex}{0ex}}f\left(x\right)=\frac{1}{x}$

The limit is $-1$

See the step by step solution

## Step 1: Given information

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

## Step 2: Calculate the average rate of change using the formula

Calculating, we get

$\underset{x\to c}{\mathrm{lim}}\left(\frac{f\left(x\right)-f\left(c\right)}{x-c}\right)\phantom{\rule{0ex}{0ex}}\underset{x\to 1}{\mathrm{lim}}\left(\frac{\frac{1}{x}-1}{x-1}\right)=\underset{x\to 1}{\mathrm{lim}}\left(-\frac{1-x}{x\left(-x+1\right)}\right)\phantom{\rule{0ex}{0ex}}\underset{x\to 1}{\mathrm{lim}}\left(\frac{\frac{1}{x}-1}{x-1}\right)==\underset{x\to 1}{\mathrm{lim}}\left(-\frac{1}{x}\right)\phantom{\rule{0ex}{0ex}}\underset{x\to 1}{\mathrm{lim}}\left(\frac{\frac{1}{x}-1}{x-1}\right)=-\frac{1}{1}\phantom{\rule{0ex}{0ex}}\underset{x\to 1}{\mathrm{lim}}\left(\frac{\frac{1}{x}-1}{x-1}\right)=-1$