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: 49


$$\lim_{x\to(\pi/2)^-}\tan x=\infty$$

Work Step by Step

$$A=\lim_{x\to(\pi/2)^-}\tan x=\lim_{x\to(\pi/2)^-}\frac{\sin x}{\cos x}$$ As $x\to(\pi/2)^-$, $\sin x$ approaches $\sin(\pi/2)=1\gt0$, while $\cos x$ approaches $\cos(\pi/2)=0$ from the left, where $\cos x\gt0$. (The left side of $\pi/2$ are the values of $x=\pi/3, \pi/4$, etc. And as we consider only values close $\pi/2$, all these values of $x$ are positive) Since as $x\to(\pi/2)^-$, $\sin x\gt0$ and $\cos x\gt0$, so $\sin x/\cos x$ will approach $\infty$. Therefore, $$A=\infty$$
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.