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: 51

Answer

\[ = \frac{a}{b}\]

Work Step by Step

\[\begin{gathered} \mathop {\lim }\limits_{x \to 0} \,\,\frac{{\sin ax}}{{\sin bx}} \hfill \\ \hfill \\ rewrite \hfill \\ \hfill \\ = \,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{bx}}{{\sin bx}}} \right)\,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{ax}}} \right)\,\left( {\frac{a}{b}} \right) \hfill \\ \hfill \\ \frac{a}{b} = \,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{bx}}{{\sin bx}}} \right)\,\left( {\mathop {\lim }\limits_{x \to 0} \frac{{\sin ax}}{{ax}}} \right)\, \hfill \\ \hfill \\ evaluate\,\,the\,\,special\,limits \hfill \\ \hfill \\ = \,\left( 1 \right)\,\left( 1 \right)\,\left( {\frac{a}{b}} \right) \hfill \\ \hfill \\ {\text{Simplify}} \hfill \\ \hfill \\ = \frac{a}{b} \hfill \\ \end{gathered} \]
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