Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 38

$\frac{1}{3}$

Work Step by Step

\begin{align*} \lim _{z\rightarrow 0} \frac{\sin (z/3)}{\sin z} &=\lim _{z\rightarrow 0} \frac{\sin (z/3)}{z} \frac{z}{\sin z} \\ &= \lim _{z\rightarrow 0} \frac{1}{3} \frac{\sin (z/3)}{z/3} \frac{z}{\sin z} \\ &=\frac{1}{3}\lim _{z/3\rightarrow 0} \frac{\sin (z/3)}{z/3}\lim _{z\rightarrow 0} \frac{z}{\sin z} \\ &=\frac{1}{3}. \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1.$

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