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: 52


$\lim_{\theta\to0}(2-\cot\theta)$ does not exist.

Work Step by Step

$$A=\lim_{\theta\to0}(2-\cot\theta)=2-\lim_{\theta\to0}\cot\theta=2-\lim_{\theta\to0}\frac{\cos\theta}{\sin\theta}$$ As $\theta\to0^+$, $\cos\theta$ approaches $\cos0=1\gt0$, while $\sin\theta$ approaches $\sin\theta=0$ from the right, where $\sin\theta\gt0$. As $\theta\to0^-$, $\cos\theta$ approaches $\cos0=1\gt0$, while $\sin\theta$ approaches $\sin\theta=0$ from the left, where $\sin\theta\lt0$. Therefore, as $\theta\to0^+$, $\cos\theta/\sin\theta$ will approach $\infty$; while as $\theta\to0^-$, $\cos\theta/\sin\theta$ will approach $-\infty$. In other words, $$\lim_{\theta\to0^+}\frac{\cos\theta}{\sin\theta}=\infty\hspace{1cm}\lim_{\theta\to0^-}\frac{\cos\theta}{\sin\theta}=-\infty$$ Since $\lim_{\theta\to0^+}\frac{\cos\theta}{\sin\theta}\ne\lim_{\theta\to0^-}\frac{\cos\theta}{\sin\theta}$, $\lim_{\theta\to0}\frac{\cos\theta}{\sin\theta}$ does not exist. So $\lim_{\theta\to0}(2-\cot\theta)$ does not exist.
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