Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.5 Derivatives of Trigonometric Functions - 3.5 Exercises: 13

Answer

\[\begin{gathered} = 7 \hfill \\ \hfill \\ \end{gathered} \]

Work Step by Step

\[\begin{gathered} \mathop {\lim }\limits_{x \to \,0} \,\,\,\,\frac{{\tan \,\,7x}}{{\sin x}} \hfill \\ \hfill \\ use\,\,the\,\,identity\,\,\tan \theta = \frac{{\sin \theta }}{{\cos \theta }} \hfill \\ \,\,\tan \,\,7x = \frac{{\sin 7x}}{{\cos 7x}} \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ = \mathop {\lim }\limits_{x \to \,0} \,\frac{{\frac{{\sin 7x}}{{\cos 7x}}}}{{\sin x}}\, = \,\,\mathop {\lim }\limits_{x \to \,0} \,\,\left( {\frac{{\sin 7x}}{{\cos 7x}}} \right)\,\left( {\frac{1}{{\sin x}}} \right) \hfill \\ \hfill \\ use\,\,the\,\,product\,\,of\,\,limits\,law \hfill \\ \hfill \\ = \,\left( {\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\frac{1}{{\cos 7x}}} \right)\,\left( {\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\,\sin 7x} \right)\,\left( {\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\frac{1}{{\sin x}}} \right) \hfill \\ \hfill \\ = \,\left( {\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\,\frac{1}{{\cos 7x}}} \right)\,\left( 7 \right)\left( {\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\,\frac{{\sin 7x}}{{7x}}} \right)\,\left( {\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\,\frac{x}{{\sin x}}} \right) \hfill \\ \hfill \\ from\,\,the\,\,theorem\,\,7.11 \hfill \\ \hfill \\ \mathop {\lim }\limits_{x \to \,0} \,\,\,\,\,\frac{{\sin 7x}}{{7x}} = 1{\text{ and }}\mathop {\lim }\limits_{x \to \,0} \,\,\,\,\,\frac{x}{{\sin x}} = 1 \hfill \\ \hfill \\ \hfill \\ = \,\left( 1 \right)\,\left( 7 \right)\,\left( 1 \right)\,\left( 1 \right) \hfill \\ \hfill \\ = 7 \hfill \\ \end{gathered} \]
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