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\begin{frame}
\frametitle{Limits at Infinity}

\begin{block}{}
Let $f$ be a function defined on some interval $(a,\infty)$.
Then
\begin{gather*}
\lim_{x\to \infty} f(x) = L\\[1ex]
\text{spoken: the limit of $f(x)$, as $x$ approaches infinity, is $L$''}
\end{gather*}
if the values $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently large.
\end{block}

\begin{center}
\scalebox{.7}{
\begin{tikzpicture}[default,xscale=.8]
\diagram{-.5}{4}{-.5}{3}{1}
\diagramannotatez
\draw[cgreen,ultra thick] plot[smooth,domain=0:4.5,samples=20,xshift=-.5cm] function{1.5- 1.5*(2*x**2 - x)/(.5+ abs(x**4))};
\draw[cred,dashed] (-.5,1.5) -- (4,1.5);
\begin{tikzpicture}[default,xscale=.8]
\diagram{-.5}{4}{-.5}{3}{1}
\diagramannotatez
\draw[cgreen,ultra thick] plot[smooth,domain=-.5:4,samples=20] function{1.5 + 2/(3*x+3)};
\draw[cred,dashed] (-.5,1.5) -- (4,1.5);
\end{tikzpicture}