\begin{align*}

\lim\limits_{x\to 3}\left(\frac{\sqrt{3x}}{x\sqrt{x+1}}\right)&=\frac{\sqrt{3\left(3\right)}}{3\sqrt{3+1}} \\
\\
&=\frac{\sqrt{9}}{3\sqrt{4}} \\
\\
& =\frac{3}{3\cdot 2}\\
\\
& =\frac{3}{6}\\
\\
&=\frac{1}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\
\\
\end{align*}