\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);
    \end{tikzpicture}~\quad~
    \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}
    ~\quad~
    \begin{tikzpicture}[default,xscale=.8]
      \diagram{-.5}{4}{-.5}{3}{1}
      \diagramannotatez
      \draw[cgreen,ultra thick] plot[smooth,domain=-.5:4,samples=200] function{1.5 + 5*sin(10*x)/(3*(x+1)**(1.5)+3)};
      \draw[cred,dashed] (-.5,1.5) -- (4,1.5);
    \end{tikzpicture}
    }
  \end{center}  
\end{frame}