University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 2 - Section 2.4 - One-Sided Limits - Exercises - Page 85: 23


$$\lim_{y\to0}\frac{\sin 3y}{4y}=\frac{3}{4}$$

Work Step by Step

$$A=\lim_{y\to0}\frac{\sin 3y}{4y}$$ To be able to apply Theorem 7, we need $3y$ in the denominator, instead of $4y$. So we would multiply both numerator and denominator by $3/4$: $$A=\lim_{y\to0}\frac{3/4\sin3y}{3/4\times4y}=\frac{3}{4}\lim_{y\to0}\frac{\sin 3y}{3y}$$ Take $\theta=3y$. Then as $y\to0$, $\theta\to3\times 0=0$ $$A=\frac{3}{4}\lim_{\theta\to0}\frac{\sin\theta}{\theta}$$ Apply Theorem 7: $$A=\frac{3}{4}\times1=\frac{3}{4}$$
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