# Chapter 2 - Limits - 2.6 Trigonometric Limits - Exercises - Page 77: 36

$$4$$

#### Work Step by Step

\begin{align*} \lim _{t\rightarrow 0} \frac{\tan 4t}{t\sec t} &=\lim _{t\rightarrow 0} \frac{\sin 4t}{\cos 4t}\frac{\cos t}{t} \\ &= \lim _{t\rightarrow 0} \frac{\sin 4t}{t}\frac{\cos t}{\cos 4t} \\ &= \lim _{t\rightarrow 0} 4 \frac{\sin 4t}{4t}\frac{\cos t}{\cos 4t} \\ &= 4\lim _{4t\rightarrow 0} \frac{\sin 4t}{4t} \lim _{t\rightarrow 0} \frac{\cos t}{\cos 4t} \\ &=4 \frac{\cos 0}{\cos 0}\\ &=4. \end{align*} Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1.$

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