\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}