University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Practice Exercises - Page 110: 4


a) $-\sqrt2$ b) $\sqrt2/2$ c) $\frac{1+2\sqrt2}{2}$ d) $2$ e) $1/2$ f) $-1/2$

Work Step by Step

$\lim_{x\to 0}f(x)=1/2$ and $\lim_{x\to 0}g(x)=\sqrt2$ a) $\lim_{x\to 0}(-g(x))=-(\lim_{x\to 0}g(x))=-\sqrt2$ (Composite Rule) b) $\lim_{x\to 0}(g(x)\times f(x))=\lim_{x\to 0}g(x)\times\lim_{x\to 0}f(x)=\sqrt2\times(1/2)=\sqrt2/2$ (Product Rule) c) $\lim_{x\to 0}(f(x)+g(x))=\lim_{x\to 0}f(x)+\lim_{x\to 0}g(x)=\frac{1}{2}+\sqrt2=\frac{1+2\sqrt2}{2}$ (Sum Rule) d) $\lim_{x\to 0}1/f(x)=\frac{1}{\lim_{x\to 0}f(x)}=\frac{1}{1/2}=2$ (Quotient Rule) e) $\lim_{x\to 0}(x+f(x))=\lim_{x\to 0}x+\lim_{x\to 0}f(x)=0+1/2=1/2$ (Sum Rule) f) $\lim_{x\to 0}\frac{f(x)\cos x}{x-1}=\frac{\lim_{x\to 0}f(x)\times\lim_{x\to 0}(\cos x)}{\lim_{x\to 0}x-\lim_{x\to 0}1}=\frac{1/2\times\cos0}{0-1}=\frac{1/2\times1}{-1}=-1/2$ (Quotient, Product and Difference Rule)
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