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: 38



Work Step by Step

The taylor series can be defined as: $\ln (1+x)=x-\dfrac{ x^2}{2}+\dfrac{x^3}{3}-....$ We have $\lim\limits_{x \to 2} \dfrac{x^2-4}{\ln (x-1)}=\lim\limits_{x \to 2} \dfrac{(x-2)(x+2)}{(x-2)-\dfrac{ (x-2)^2}{2}+\dfrac{(x-2)^3}{3}-....)} \\= \dfrac{ \lim\limits_{x \to 2} [(x-2)(x+2)] }{\lim\limits_{x \to 2} (x-2)-\dfrac{ (x-2)^2}{2}+\dfrac{(x-2)^3}{3}-....)} \\ =\dfrac{2+2}{1-0+0...}\\=4$
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