University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 9 - Section 9.10 - The Binomial Series and Applications of Taylor Series - Practice Exercises - Page 553: 77



Work Step by Step

The Taylor series for $\cos x $ can be defined as: $\cos x=1-\dfrac{x^2}{2!}+\dfrac{ x^4}{4!}-....$ and the Taylor series for $\sin x $ can be defined as: $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-....$ Now, $\lim\limits_{z \to 0} \dfrac{1-\cos^2 z}{\ln (1-z) +\sin z}=\lim\limits_{z \to 0} \dfrac{1- (1-z^2+z^4/3-...)}{(-z-\dfrac{z^2}{2}-....)+(z-\dfrac{z^3}{3!}+.....}$ or, $=\lim\limits_{z \to 0} \dfrac{1-\dfrac{z^2}{3}+...}{-\dfrac{1}{2}-\dfrac{z}{3}-....}$ or, $=\dfrac{1}{-1/2}$ or, $=-2$
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