\begin{frame}
  \frametitle{Properties of Limits}
  
  \begin{block}{}
    If
    \begin{itemize}
      \item $f(x) \le g(x)$ when $x$ is near $a$ (except possibly $a$),
      \item $\lim_{x\to a} f(x)$ exists, and 
      \item $\lim_{x\to a} g(x)$ exist,
    \end{itemize} 
    then
    \begin{talign}
      \lim_{x\to a} f(x) \le \lim_{x\to a} g(x)
    \end{talign}    
  \end{block}
  \pause
  Formally, near $a$ means on $(a-\epsilon,a+\epsilon) \setminus \{a\}$ for some $\epsilon > 0$.
  \pause\bigskip
  
  \begin{exampleblock}{}
    We have $x^3 \le x^2$ for $x \in (-1,1)$. 
    \pause\medskip
    
    As a consequence:
    \begin{talign}
      \lim_{x \to a} x^3 \le \lim_{x\to a} x^2
    \end{talign}
    for all $a \in (-1,1)$.
  \end{exampleblock}
\end{frame}