Answer
$$4$$
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. $
