Calculus: Early Transcendentals 8th Edition

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

Chapter 3 - Section 3.3 - Derivatives of Trigonometric Functions - 3.3 Exercises - Page 197: 48


$$A=\lim\limits_{x\to0}\frac{\sin x^2}{x}=0$$

Work Step by Step

$$A=\lim\limits_{x\to0}\frac{\sin x^2}{x}$$ $$A=\lim\limits_{x\to0}\Bigg(\frac{\sin x^2}{x^2}\times x\Bigg)$$ $$A=\lim\limits_{x\to0}\frac{\sin x^2}{x^2}\times\lim\limits_{x\to0} x$$ Let $x^2=\theta$. Then as $x\to0$, $\theta\to0$. So, $$A=\lim\limits_{\theta\to0}\frac{\sin\theta}{\theta}\times\lim\limits_{x\to0} x$$ $$A=1\times0=0$$
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