Answer
$$\frac{1}{25}$$
Work Step by Step
\begin{align*}
\lim _{x\rightarrow 0} x \csc 25x &=\lim _{x\rightarrow 0} \frac{ x}{\sin 25x} \\
&= \lim _{x\rightarrow 0} \frac{1}{25} \frac{ 25 x}{\sin 25x} \\
&= \frac{1}{25}\lim _{25x\rightarrow 0} \frac{ 25 x}{\sin 25x} \\
&= \frac{1}{25}.
\end{align*}
Where we used Theorem 2 -- that is, $\lim _{x\rightarrow 0}\frac{ \sin x}{ x}=1. $
