University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 4 - Practice Exercises - Page 277: 67



Work Step by Step

Consider $f(x)=\lim\limits_{x \to \dfrac{\pi}{2}^{-}}\sec 7 x \cos 3x=\lim\limits_{k \to 0^{+}}\dfrac{\sin 3k}{\sin 7k}=\lim\limits_{k \to 0^{+}} \dfrac{a(x)}{b(x)}$ and $a(0)=0, b(0)=0$ Thus, $f(0)=\dfrac{0}{0}$ This shows an Inderminate form of the limit, so apply L-Hospital's rule: $\lim\limits_{x \to l}\dfrac{a(x)}{b(x)}=\lim\limits_{x \to l}\dfrac{a'(x)}{b'(x)}$ Thus, $\lim\limits_{k \to 0^{+}}\dfrac{3\cos 3k}{7 \cos 7k}=\dfrac{3}{7}$
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