\begin{frame} \frametitle{Limits at Infinity} \begin{exampleblock}{} For exponential function we have: \begin{talign} \lim_{x \to \infty} a^x &= 0 &&\text{for $0 \le a < 1$}\\[1ex] \lim_{x \to -\infty} a^x &= 0 &&\text{for $a > 1$} \end{talign} \end{exampleblock} \pause\bigskip \begin{exampleblock}{} For any polynomial $P$ and $a > 1$ we have \begin{talign} \lim_{x \to \infty} \frac{P(x)}{a^x} &= 0 \end{talign} since the exponential function grows after than any polynomial. \end{exampleblock} \pause\bigskip \begin{exampleblock}{} For any polynomial $P$ and \alert{$0 < a < 1$} we have \begin{talign} \lim_{x \to \alert{-}\infty} \frac{P(x)}{a^x} &= 0 \end{talign} \end{exampleblock} \end{frame}