Thomas' Calculus 13th Edition

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

Chapter 10: Infinite Sequences and Series - Practice Exercises - Page 637: 75

Answer

$$\dfrac{1}{12}$$

Work Step by Step

The Taylor series can be written as follows: $\sin x= x-\dfrac{x^3}{3!}+\dfrac{ x^5}{5!}-...........$ and $\cos x=1-\dfrac{x^2}{2}+\dfrac{ x^3}{3}-....$ $$\lim\limits_{t \to 0} \dfrac{1}{2-2 \cos t}-\dfrac{1}{t^2}=\lim\limits_{t \to 0} \dfrac{t^2-2+2 \cos t }{2t^2 (1-\cos t)} \\=\lim\limits_{t \to 0} \dfrac{t^2-2+2 (1-\dfrac{t^2}{2!}+\dfrac{ t^4}{4}-....) }{2t^2 (1-1+\dfrac{t^2}{2!}+\dfrac{ t^4}{4}-....)} \\=\dfrac{1}{12}$$
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