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: 24



Work Step by Step

Consider: $\lim\limits_{t \to 0}f(t)=\lim\limits_{t \to 0}\dfrac{t-\sin t}{1-\cos t}$ WenNeed to check that the limit has an indeterminate form. Thus, $f(0)=\dfrac{0}{0}$ 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 $\lim\limits_{t \to 0}\dfrac{\sin t+t \cos t}{\sin t}=\dfrac{0}{0}$ WenNeed to apply L-Hospital's rule again. $\lim\limits_{t \to 0}\dfrac{2\cos t-t \sin t}{\cos t}=\dfrac{2(1)-0}{1}=2$
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