Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.5 - Continuity - 2.5 Exercises - Page 125: 30


$B(x)$ is continuous on $(-2,\frac{-\pi}{2})\cup(\frac{-\pi}{2},\frac{\pi}{2})\cup(\frac{\pi}{2},2)$.

Work Step by Step

$$B(x)=\frac{\tan x}{\sqrt{4-x^2}}$$ 1) Find the domain of the numerator The domain of $\tan x$ is $x\in R$ except where $x=\frac{\pi}{2}+k\pi (k\in Z)$. According to Theorem 7, $\tan x$ is continuous on $R$ except where $x=\frac{\pi}{2}+k\pi (k\in Z)$. 2) Find the domain of the denominator $\sqrt{4-x^2}$ is defined for $$4-x^2\geq0$$$$x^2\le4$$$$-2\le x\le2$$ Therefore, the domain of $\sqrt{4-x^2}$ is $x\in[-2,2]$. According to Theorem 7, $\sqrt{4-x^2}$ is continuous on $[-2,2]$. 3) Combining the domain in 1) and 2), we see that $\frac{\pi}{2}$ and $\frac{-\pi}{2}$ are within $[-2,2]$. Therefore, we must exclude them out of the continuity range. So, $\tan x$ and $\sqrt{4-x^2}$ are both continuous on $[-2,\frac{-\pi}{2})\cup(\frac{-\pi}{2},\frac{\pi}{2})\cup(\frac{\pi}{2},2]$. According to part 5) of Theorem 4, $B(x)$ is also continuous on $[-2,\frac{-\pi}{2})\cup(\frac{-\pi}{2},\frac{\pi}{2})\cup(\frac{\pi}{2},2]$ except where $$\sqrt{4-x^2}=0$$$$4-x^2=0$$$$x^2=4$$$$\left\{\begin{array} {c l}x=2\\x=-2\end{array}\right.$$ Therefore, $B(x)$ is continuous on $(-2,\frac{-\pi}{2})\cup(\frac{-\pi}{2},\frac{\pi}{2})\cup(\frac{\pi}{2},2)$.
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