\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}{}
    Find $\lim_{x\to 0} |x|$ where
    \begin{talign}
      |x| = \begin{cases}
        x&\text{for $x \ge 0$}\\
        -x&\text{for $x < 0$}
      \end{cases}
    \end{talign}
    \pause
    Since $|x| = x$ for all $x > 0$ we obtain:
    \begin{talign}
      \lim_{x\to 0^+} |x| = \lim_{x\to 0^+} x \mpause[1]{ = 0} 
    \end{talign}
    \pause\pause
    Since $|x| = -x$ for all $x < 0$ we obtain:
    \begin{talign}
      \lim_{x\to 0^-} |x| = \lim_{x\to 0^-} -x \mpause[1]{ = 0} 
    \end{talign}
    \pause\pause
    Hence $\lim_{x\to 0} |x| = 0$.
  \end{exampleblock}
  \vspace{10cm}
\end{frame}