Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.2 Algebra Techniques for Finding Limits - 14.2 Assess Your Understanding - Page 884: 51



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: $f(x)=\dfrac{1}{x}$ 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}-1}{x-1} \\=\lim\limits_{x\to 1}\dfrac{\dfrac{1-x}{x}}{x-1} \\=\dfrac{\lim\limits_{x\to 1}-(x-1)}{\lim\limits_{x\to 1} x(x-1)} \\=\lim\limits_{x\to 1} (-\dfrac{1}{x} ) \\=\dfrac{-1}{1} \\=-1 $
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