Calculus: Early Transcendentals (2nd Edition)

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

Chapter 2 - Limits - 2.5 Limits at Infinity - 2.5 Exercises - Page 97: 72

Answer

$${\text{200}}$$

Work Step by Step

$$\eqalign{ & {\text{Let }}m\left( t \right) = 200\left( {1 - {2^{ - t}}} \right) \cr & {\text{Calculate }}\mathop {\lim }\limits_{t \to \infty } m\left( t \right) \cr & {\text{ }}\mathop {\lim }\limits_{t \to \infty } m\left( t \right) = \mathop {\lim }\limits_{t \to \infty } 200\left( {1 - {2^{ - t}}} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 200\mathop {\lim }\limits_{t \to \infty } \left( {1 - {2^{ - t}}} \right) \cr & {\text{Evaluate the limit}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 200\left[ {\mathop {\lim }\limits_{t \to \infty } \left( 1 \right) - \overbrace {\mathop {\lim }\limits_{t \to \infty } {2^{ - t}}}^{{\text{approaches to 0}}}} \right] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 200\left( {1 - 0} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 200 \cr & {\text{Therefore}}{\text{, }} \cr & {\text{The steady state exits}} \cr & {\text{The steady - state value is 200}} \cr} $$
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