Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 3


$$\lim _{x \rightarrow 7} f(x)=6$$

Work Step by Step

The Squeeze Theorem does not require that the inequalities $$l(x) \leq f(x) \leq u(x)$$ hold for all $x,$ but only that the inequalities hold on some open interval containing $x=c .$ From the given figure, it is clear that $$l(x) \leq f(x) \leq u(x)$$ on some open interval containing $x=7 .$ Because $$\lim _{x \rightarrow 7} u(x)=\lim _{x \rightarrow 7} l(x)=6,$$ the Squeeze Theorem guarantees that $$\lim _{x \rightarrow 7} f(x)=6$$
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