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 549: 29



Work Step by Step

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