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: 65



Work Step by Step

Consider $f(x)=\lim\limits_{x \to 0}\dfrac{\sin^2 x}{\tan x^2}=\lim\limits_{x \to 0} \dfrac{a(x)}{b(x)}$ 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, $\lim\limits_{x \to 0}\dfrac{2 \sin x \cos x}{2x \sec^2 x^2}=\lim\limits_{x \to 0}\dfrac{\sin x}{x} \cdot \lim\limits_{x \to 0}\dfrac{\cos x}{\sec^2 x^2}=1 \cdot 1=1$
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