Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.5 Derivatives of Trigonometric Functions - 3.5 Exercises: 50


$\lim_{x\to0}\dfrac{\sin ax}{bx}=\dfrac{a}{b}$

Work Step by Step

$\lim_{x\to0}\dfrac{\sin ax}{bx}$ $;$ $a$ and $b$ are constants and $b\ne0$ Divide both the numerator and the denominator by $ax$: $\lim_{x\to0}\dfrac{\sin ax}{bx}=\lim_{x\to0}\dfrac{\dfrac{\sin ax}{ax}}{\dfrac{bx}{ax}}=\lim_{x\to0}\dfrac{\dfrac{\sin ax}{ax}}{\Big(\dfrac{b}{a}\Big)}=...$ Use the quotient limit law to evaluate the limit: $...=\dfrac{\lim_{x\to0}\dfrac{\sin ax}{ax}}{\lim_{x\to0}\dfrac{b}{a}}=\dfrac{1}{\dfrac{b}{a}}=\dfrac{a}{b}$
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