Thomas' Calculus 13th Edition

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

Chapter 7: Transcendental Functions - Section 7.5 - Indeterminate Forms and L'Hopital's Rule - Exercises 7.5 - Page 410: 75


a) The limit provided in part(a) is not correct. b) The limit provided in part(b) is correct.

Work Step by Step

(a) L-Hospital's rule is defined as $\lim\limits_{x \to \infty} f(x)=\lim\limits_{x \to \infty} \dfrac{p'(x)}{q'(x)}$ Here, $\lim\limits_{x \to 3} \dfrac{x-3}{x^2-3}=\lim\limits_{x \to 3} \dfrac{1}{2x}=\dfrac{1}{6}$ Also, $f(3)=\dfrac{3-3}{3^2-3}=\dfrac{0}{6} \ne \dfrac{0}{0}$ Here, we cannot use L-Hospital's rule because we do not have any indeterminate form (b) $\lim\limits_{x \to 3} \dfrac{x-3}{x^2-3}=\dfrac{3-3}{3^2-3}$ or, $\dfrac{0}{6}=0$
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