Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.2 Limits: A Numerical and Graphical Approach - Exercises - Page 54: 23


$\displaystyle\lim_{\theta\rightarrow 0} \dfrac{\cos\theta-1}{\theta}=0$

Work Step by Step

We have to estimate the limit: $\displaystyle\lim_{\theta\rightarrow 0} \dfrac{\cos\theta-1}{\theta}$ Compute $\dfrac{\cos\theta-1}{\theta}$ for values of $\theta$ close to 0: $\dfrac{\cos (-0.1)-1}{-0.1}\approx 0.0499583$ $\dfrac{\cos (-0.01)-1}{-0.01}\approx 0.0049996$ $\dfrac{\cos (-0.001)-1}{-0.001}\approx 0.0005$ $\dfrac{\cos (-0.0001)-1}{0.0001}\approx 0.00005$ $\dfrac{\cos (0.0001)-1}{0.0001}\approx 0.00005$ $\dfrac{\cos (0.001)-1}{0.001}\approx 0.0005$ $\dfrac{\cos (0.01)-1}{0.01}\approx 0.0049996$ $\dfrac{\cos (0.1)-1}{0.1}\approx 0.0499583$ Therefore we have: $\displaystyle\lim_{\theta\rightarrow 0} \dfrac{\cos\theta-1}{\theta}=0$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.