Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - Review Exercises: 24

Answer

$\lim\limits_{s\to0}\dfrac{[1/\sqrt{1+s}]-1}{s}=-\dfrac{1}{2}.$

Work Step by Step

$\lim\limits_{s\to0}\dfrac{[1/\sqrt{1+s}]-1}{s}=\lim\limits_{s\to0}\dfrac{1-\sqrt{1+s}}{s\sqrt{1+s}}$ $=\lim\limits_{s\to0}[\dfrac{1-\sqrt{s+1}}{s\sqrt{s+1}}\times\dfrac{1+\sqrt{s+1}}{1+\sqrt{s+1}}]$ $=\lim\limits_{s\to0}\dfrac{(1)^2-(\sqrt{s+1})^2}{s\sqrt{s+1}(1+\sqrt{s+1})}=\lim\limits_{s\to0}\dfrac{-1}{\sqrt{s+1}(1+\sqrt{s+1})}$ $=\dfrac{-1}{\sqrt{1+0}(1+\sqrt{0+1})}=-\dfrac{1}{2}.$
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