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: 35



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 \infty} x^2 (e^{1/x^2}-1)=\lim\limits_{x \to \infty} x^2 (-1+e^{1/x^2}) $ or, $=\lim\limits_{x \to \infty} x^2 [-1+1-\dfrac{1}{x^2}+\dfrac{1}{2x^4}-\dfrac{1}{6x^6}+.....]$ or, $=\lim\limits_{x \to \infty} -1+\dfrac{1}{2x^2}-\dfrac{1}{6x^4}+....$ or, $=-1$
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