Calculus 10th Edition

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

Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.7 Exercises - Page 564: 1

Answer

$$\frac{4}{3}$$

Work Step by Step

\[\begin{gathered} \mathop {\lim }\limits_{x \to 0} \frac{{\sin 4x}}{{\sin 3x}} \hfill \\ {\text{Evaluate }}f\left( x \right){\text{ for the given values and complete the table}}{\text{.}} \hfill \\ x = - 0.1 \to f\left( { - 0.1} \right) = \frac{{\sin \left[ {4\left( { - 0.1} \right)} \right]}}{{\sin \left[ {3\left( { - 0.1} \right)} \right]}} \approx 1.3177 \hfill \\ x = - 0.01 \to f\left( { - 0.01} \right) = \frac{{\sin \left[ {4\left( { - 0.01} \right)} \right]}}{{\sin \left[ {3\left( { - 0.01} \right)} \right]}} \approx 1.3332 \hfill \\ x = - 0.001 \to f\left( { - 0.001} \right) = \frac{{\sin \left[ {4\left( { - 0.001} \right)} \right]}}{{\sin \left[ {3\left( { - 0.001} \right)} \right]}} \approx 1.3333 \hfill \\ x = 0.001 \to f\left( {0.001} \right) = \frac{{\sin \left[ {4\left( {0.001} \right)} \right]}}{{\sin \left[ {3\left( {0.001} \right)} \right]}} \approx 1.3333 \hfill \\ x = 0.01 \to f\left( {0.01} \right) = \frac{{\sin \left[ {4\left( {0.01} \right)} \right]}}{{\sin \left[ {3\left( {0.01} \right)} \right]}} \approx 1.3332 \hfill \\ x = 0.1 \to f\left( {0.1} \right) = \frac{{\sin \left[ {4\left( {0.1} \right)} \right]}}{{\sin \left[ {3\left( {0.1} \right)} \right]}} \approx 1.3177 \hfill \\ \boxed{\begin{array}{*{20}{c}} x&{ - 0.1}&{ - 0.01}&{ - 0.001}&{0.001}&{0.01}&{0.1} \\ {f\left( x \right)}&{1.3177}&{1.3332}&{1.3333}&{1.3333}&{1.3332}&{1.3177} \end{array}} \hfill \\ {\text{Therefore,}} \hfill \\ \mathop {\lim }\limits_{x \to 0} \frac{{\sin 4x}}{{\sin 3x}} \approx 1.3333 \hfill \\ {\text{Exact value :}} \hfill \\ \mathop {\lim }\limits_{x \to 0} \frac{{\sin 4x}}{{\sin 3x}} = \frac{4}{3} \hfill \\ {\text{Graph}} \hfill \\ \end{gathered} \]
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