Calculus (3rd Edition)

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

Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 41



Work Step by Step

\begin{align*} \lim _{t\rightarrow 0} \frac{\csc 8t}{\csc 4t} &=\lim _{t\rightarrow 0} \frac{\sin 4t}{\sin 8t} \\ &= \lim _{t\rightarrow 0} \frac{4}{8} \frac{\sin 4t}{4t} \frac{8t}{\sin 8t} \\ &= \frac{4}{8}\lim _{4t\rightarrow 0} \frac{\sin 4t}{4t} \lim _{8t\rightarrow 0}\frac{8t}{\sin 8t} \\ &=\frac{4}{8}=\frac{1}{2}. \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1. $
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.