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 - Section 10.10 - The Binomial Series and Applications of Taylor Series - Exercises 10.10 - Page 633: 30



Work Step by Step

$\lim\limits_{x \to 0}\dfrac{e^x-e^{-x}}{x}=\lim\limits_{x \to 0} \dfrac{[(1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+....)-(1-x+\dfrac{x^2}{2!}-\dfrac{x^3}{3!}+\dfrac{x^4}{4!}-....)]}{x}\\=\lim\limits_{x \to 0} \dfrac{(2x)+2 \times \dfrac{x^3}{3!}+2 \times \dfrac{x^5}{5!}+...}{x}\\=\lim\limits_{x \to 0} (2)+\lim\limits_{x \to 0} \dfrac{2x^3}{3!}+\lim\limits_{x \to 0} \dfrac{2x^4}{4!}+....]\\=2$
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