University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 2 - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises - Page 108: 51



Work Step by Step

$$A=\lim_{\theta\to0^-}(1+\csc\theta)=1+\lim_{\theta\to0^-}\csc\theta=1+\lim_{\theta\to0^-}\frac{1}{\sin\theta}$$ As $\theta\to0^-$, $\sin x$ approaches $\sin0=0$ from the left, where $\sin\theta\lt0$. ($\theta\to0^-$ are values of $\theta$ like $-\pi/6, -\pi/4$, etc. and all these values give $\sin \theta$ negative values) Therefore, $1/\sin \theta$ will approach $-\infty$. In other words, $$A=1+(-\infty)=-\infty$$
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