Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.4 - The Precise Definition of a Limit - 2.4 Exercises - Page 113: 10

Answer

(a) $\delta = 0.04$ (b) $\delta = 0.03$
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Work Step by Step

Note that the graph has symmetry about the line $x = \pi$ According to Definition 6, if $\lim\limits_{x \to \pi}csc^2~x = \infty$, then for any positive number $M$, there is a number $\delta$ such that if $\vert x-\pi \vert \lt \delta$, then $csc^2~x \gt M$ (a) On the graph, we can see that when $3.10 \lt x \lt 3.18$, then $csc^2 ~x \gt 500$ Thus, when $\vert x-\pi \vert \lt 0.04$, then $csc^2~x \gt 500$ $\delta = 0.04$ (b) On the graph, we can see that when $3.11 \lt x \lt 3.17$, then $csc^2 ~x \gt 1000$ Thus, when $\vert x-\pi \vert \lt 0.03$, then $csc^2~x \gt 1000$ $\delta = 0.03$
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