{
   \begin{eqnarray*}
      \lim_{x\to1}\frac{x^3-3x+2}{x^4-4x+3}
      & = &\lim_{x\to1}\frac{{(x-1)}^2(x+2)}{{(x-1)}^2(x^2+2x+3)}\\
      & = &{\left(\lim_{x\to1}\frac{x-1}{x-1}\right)}^2
            \cdot\lim_{x\to1}\frac{x+2}{x^2+2x+3}\;=\;\frac12.
   \end{eqnarray*}
   }