\begin{frame}
  \frametitle{Computing Limits: Function Replacement}
  
  For one-sided limits we have:  
  \begin{block}{}
    If $f(x) = g(x)$ for all $x < a$, then $\lim_{x \to a^-} f(x) = \lim_{x \to a^-} g(x)$.
  \end{block}
  \begin{block}{}
    If $f(x) = g(x)$ for all $x > a$, then $\lim_{x \to a^+} f(x) = \lim_{x \to a^+} g(x)$.
  \end{block}
  \bigskip
  
  \begin{exampleblock}{}
    Proof that $\lim_{x\to 0} \frac{|x|}{x}$ does not exist. 
    \pause\bigskip
    
    For all $x > 0$ we have $\frac{|x|}{x} \pause = \frac{x}{x}\pause = 1$. \pause Thus
    \begin{talign}
      \lim_{x\to 0^+} \frac{|x|}{x} = \lim_{x\to 0^+} 1 = 1
    \end{talign}
    \pause
    For all $x < 0$ we have $\frac{|x|}{x} \pause = \frac{-x}{x}\pause = -1$. \pause Thus
    \begin{talign}
      \lim_{x\to 0^-} \frac{|x|}{x} = \lim_{x\to 0^-} -1 = -1
    \end{talign}
    \pause
    Hence $\lim_{x\to 0} \frac{|x|}{x}$ does not exist since 
    $\lim_{x\to 0^-} \frac{|x|}{x} \ne \lim_{x\to 0^+} \frac{|x|}{x}$.
  \end{exampleblock}
  \vspace{10cm}
\end{frame}