University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 4 - Section 4.5 - Indeterminate Forms and L'Hôpital's Rule - Exercises - Page 248: 11



Work Step by Step

Consider: $\lim\limits_{x \to \infty}f(x)=\lim\limits_{x \to \infty}\dfrac{5x^3-2x}{7x^3+3}$ Now, $f(\infty)=\dfrac{\infty}{\infty}$ The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$ Thus: $\lim\limits_{x \to \infty}\dfrac{15x^2-2}{21x^2}=\dfrac{\infty}{\infty}$ We need to again apply L-Hospital's rule. $\lim\limits_{x \to \infty}\dfrac{30x}{42x}=\dfrac{\infty}{\infty}$ Now, again apply L-Hospital's rule. Thus, $\lim\limits_{x \to \infty}\dfrac{30}{42}=\dfrac{5}{7}$
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