{
   \begin{eqnarray*}
      \lim_{x\to1}\frac{x^{100}-2x+1}{x^{50}-2x+1}
      & = & \lim_{x\to1}\frac{x^{100}-1-2x+2}{x^{50}-1-2x+2}         \\
      & = & \lim_{x\to1}\frac{(x-1)(x^{99}+x^{98}+\ldots+x+1)-2(x-1)}
                             {(x-1)(x^{49}+x^{48}+\ldots+x+1)-2(x-1)}\\
      & = & \lim_{x\to1}\frac{(x-1)(x^{99}+x^{98}+\ldots+x-1)}
                             {(x-1)(x^{49}+x^{48}+\ldots+x-1)}       \\
      & = & \lim_{x\to1}\frac{x-1}{x-1}\cdot
            \lim_{x\to1}\frac{x^{99}+x^{98}+\ldots+x-1}
                             {x^{49}+x^{48}+\ldots+x-1}              \\
      & = & \lim_{x\to1}\frac{x^{99}+\ldots+x^{50}}
                                 {x^{49}+\ldots+x-1}
            + \lim_{x\to1}\frac{x^{49}+\ldots+x-1}
                                 {x^{49}+\ldots+x-1}                 \\
      & = & 1 + \frac{\lim_{x\to1}(x^{99}+\ldots+x^{50})}
                     {\lim_{x\to1}(x^{49}+\ldots+x-1)}               \\
      & = & 1 + \frac{50}{48}\;=\;2\frac1{24}.
   \end{eqnarray*}
   }