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: 18



Work Step by Step

Consider: $\lim\limits_{\theta \to \dfrac{-\pi}{3}}f(\theta)=\lim\limits_{ \theta \to \dfrac{-\pi}{3}}\dfrac{3 \theta +\pi}{\sin (\theta+\dfrac{\pi}{3})}$ Now, $f(\dfrac{-\pi}{3})=\dfrac{3 (\dfrac{-\pi}{3}) +\pi}{\sin ((\dfrac{-\pi}{3})+\dfrac{\pi}{3})}=\dfrac{0}{0}$ The limit shows an indeterminate form. Thus, apply L-Hospital's rule: $\lim\limits_{a \to b}f(x)=\lim\limits_{a \to b}\dfrac{g'(x)}{h'(x)}$ Then: $\lim\limits_{\theta \to \dfrac{-\pi}{3}}\dfrac{3}{\cos (\theta+\dfrac{\pi}{3})}=\dfrac{3}{\cos (\dfrac{-\pi}{3}+\dfrac{\pi}{3})}=\dfrac{3}{\cos 0}=\dfrac{3}{1}=3 $
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