Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - 1.4 Exercises - Page 80: 71

Answer

The function $h(x)=f(g(x))= \tan \frac{x}{2}$ is continuous everywhere except at the points$$\mathbb{R}- \{ (2n+1) \pi \mid n \in \mathbb{Z} \}.$$

Work Step by Step

As we know, the trigonometric functions $\sin x$ and $\cos x$ are continuous everywhere, so the function $h(x)=f(g(x))= \tan \frac{x}{2}=\frac{ \sin \frac{x}{2}}{\cos \frac{x}{2}}$ is continuous everywhere except at those points vanishing the denominator. So the discontinuities of the function are$$\cos \frac{x}{2}=0 \quad \Rightarrow \quad x= \{ (2n+1) \pi \mid n \in \mathbb{Z} \}$$Thus, the function $h(x)=f(g(x))= \tan \frac{x}{2}$ is continuous at the other points, $\mathbb{R}- \{ (2n+1) \pi \mid n \in \mathbb{Z} \}$, according to Theorem 1.12, Continuity of a Composite Function.
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