Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.2 - Limit of a Function and Limit Laws - Exercises 2.2 - Page 57: 61



Work Step by Step

$\lim\limits_{h\to0}\dfrac{f(x+h)f(x)}{h}=\lim\limits_{h\to0}\dfrac{h}{h(\sqrt{x+h}+\sqrt x)}=\lim\limits_{h\to0}\dfrac{1}{(\sqrt{x+h}+\sqrt x)}$ Plug $x=7$ we get $\lim\limits_{h\to0}\dfrac{f(x+h)-f(x)}{h}=\lim\limits_{h\to0}\dfrac{1}{\sqrt{7+h}+\sqrt7}=\dfrac{1}{\sqrt7+\sqrt7}$ and $\lim\limits_{h\to0}\dfrac{f(x+h)-f(x)}{h}=\dfrac{1}{2\sqrt7}$ Multiply both numerator and denominator with $\sqrt7$, we have $\lim\limits_{h\to0}\dfrac{f(x+h)-f(x)}{h}=\dfrac{\sqrt7}{2\times7}=\dfrac{\sqrt7}{14}$
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