\begin{align*}

\lim_{x\to 2}\left(x^2-4x+3\right)& = \lim_{x\to 2}\left(x^2\right)-\lim_{x\to 2}\left(4x\right)+\lim_{x\to 2}\left(3\right)\\

& =\left[\lim_{x\to 2}\left(x\right)\right]^2-4\lim_{x\to 2}\left(x\right)+3\\

& =\left(2\right)^2-4\left(2\right)+3\\

& =-1 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)\\

\end{align*}