calculus:resources:calculus_flipped_resources:limits:1.5_limit_tex [Department of Mathematics and Statistics, Binghamton University]

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TeX code compiled with \documentclass{beamer} using the Amsterdam theme.\\ \\ 

<nowiki>
\begin{document}
\begin{frame}
    \large The statement ”Whether or not $\displaystyle\lim_{x\rightarrow a} f(x)$ exists, 
    depends on how $f(a)$ is defined,” is true
        \begin{itemize}
            \item[(a)] sometimes,
            \item[(b)] always,
            \item[(c)] never.
        \end{itemize}
\end{frame}

\begin{frame}
    \Large Find the following limits.
    \vskip 15pt

    \begin{columns}
    \begin{column}{0.5\textwidth}
    \begin{enumerate}
    \item[\bf a)] $\displaystyle\lim_{x\to 7^-}\displaystyle\frac{x+6}{x-7}$
    \vskip 30pt

    \item[\bf b)] $\displaystyle\lim_{x\to 4}\displaystyle\frac{3-x}{(x-4)^2}$
     \end{enumerate}

    \end{column}

    \begin{column}{0.5\textwidth}
 \begin{enumerate}
    \item[\bf c)] $\displaystyle\lim_{x\to 1^+}\displaystyle\frac{8}{x^3-1}$
    \vskip 30pt

    \item[\bf d)] $\displaystyle\lim_{x\to 1^-}\displaystyle\frac{8}{x^3-1}$
    \vskip 30pt
    \end{enumerate}
    \end{column}
    \end{columns}
\end{frame}

\begin{frame}
\LARGE
If a function $f$ is not defined at $x=a$,
\begin{enumerate}[a)]
\item $\displaystyle{\lim_{x\rightarrow a} f(x)}$ cannot exist
\item $\displaystyle{\lim_{x\rightarrow a} f(x)}$ could be $0$
\item $\displaystyle{\lim_{x\rightarrow a} f(x)}$ must approach $\infty$
\item none of the above.
\end{enumerate}
\end{frame}

\begin{frame}
      \Large Draw the graph of a function $f(x)$ such that $\displaystyle\lim_{x\to 4} f(x)=5$ 
      and $f(4)=5$, or explain why this is impossible.
      \vskip 30pt

      Draw the graph of a function $g(x)$ such that $\displaystyle\lim_{x\to 4} g(x)=5$ 
      and $g(4)=4$, or explain why this is impossible.
      \vskip 30pt

         Draw the graph of a function $h(x)$ such that $\displaystyle\lim_{x\to 4} h(x)=5$ 
         and $h(4)$ is undefined, or explain why this is impossible.
\end{frame}


\begin{frame}
     \Large Draw the graph of a function $f(x)$ such that $\displaystyle\lim_{x\to 6^-} f(x)=5$ 
     and $\displaystyle\lim_{x\to 6^+} f(x)=7$, or explain why this is impossible.
      \vskip 30pt

      Draw the graph of a function $g(x)$ such that $\displaystyle\lim_{x\to 6^-} g(x)=5$ and $\displaystyle\lim_{x\to 6^+} g(x)=7$ and $g(6)=10$, or explain why this is impossible.
      \vskip 30pt

      Draw the graph of a function $h(x)$ such that $\displaystyle\lim_{x\to 6^-} g(x)=5$ 
      and $\displaystyle\lim_{x\to 6^+} g(x)=5$ and $\dlim_{x\to 6} g(x)$ is undefined, 
      or explain why this is impossible.
\end{frame}

\begin{frame}
    If all that you know about a function $g(x)$ is that $g(5)=-3$ and $g'(5)=4$,
    what is your best estimate of $g(7)$?
\end{frame}

\end{document}
</nowiki>