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 - Exercises - Page 550: 37



Work Step by Step

Taylor series for $\cos x $ can be defined as: $\cos x=1-\dfrac{ x^2}{2!}+\dfrac{x^4}{4!}-....$ Now, $\lim\limits_{x \to 0} \dfrac{\ln (1+x^2)}{1-\cos x}=\lim\limits_{x \to 0} \dfrac{(x^2-x^4/2+x^6/3)}{1-(1-\dfrac{ x^2}{2!}+\dfrac{x^4}{4!}-....)}$ or, $= \dfrac{\lim\limits_{x \to 0}(x^2-x^4/2+x^6/3)}{\lim\limits_{x \to 0}[1-(1-\dfrac{ x^2}{2!}+\dfrac{x^4}{4!}-....)]}$ or, $=2 !$ or, $=2$
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