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 409: 11



Work Step by Step

Consider: $\lim\limits_{x \to \infty}f(x)=\lim\limits_{x \to \infty}\dfrac{5x^3-2x}{7x^3+3}$ Need to check that the limit has an indeterminate form. Thus, $f(\infty)=\dfrac{\infty}{\infty}$ Now, apply L-Hospital's rule such as: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$ This implies: $\lim\limits_{x \to \infty}\dfrac{15x^2-2}{21x^2}=\dfrac{\infty}{\infty}$ Again apply L-Hospital's rule. $\lim\limits_{x \to \infty}\dfrac{30x}{42x}=\dfrac{\infty}{\infty}$ Now, again apply L-Hospital's rule. $\lim\limits_{x \to \infty}\dfrac{30}{42}=\dfrac{5}{7}$
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