Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 98: 49



Work Step by Step

$\displaystyle \tan x=\frac{\sin x}{\cos x}$ You can sketch a unit circle. 0 radians is at (1,0); the positive direction is counterclockwise. $\displaystyle \frac{\pi}{2}$ radians is positioned at $(0,1).$ When $x\displaystyle \rightarrow(\frac{\pi}{2})^{-}$, x is a radian angle measure in the first quadrant, (slightly less than $\displaystyle \frac{\pi}{2} $is on the clockwise side of $\displaystyle \frac{\pi}{2}$) where both sine and cosine are positive. The numerator, $\sin x\rightarrow 1$, (positive) and the denominator, $\cos x\rightarrow 0$, (positive). Thus, $\displaystyle \frac{\sin x}{\cos x}=\tan x\rightarrow+\infty$
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