\begin{align*}

\lim\limits_{x\to 0}\left(\frac{3x+2}{x^2-2x+4}\right)& =\frac{3\left(0\right)+2}{\left(0\right)^2-2\left(0\right)+4}\\
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& =\frac{0+2}{0-0+4}\\
\\
& =\frac{2}{4}\\
\\
& =\frac{1}{2} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\
\\

\end{align*}