\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}{Definition} \newtheorem{example}{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.6in,right=0.6in,top=0.6in,bottom=0.6in} \pagestyle{fancy} \lhead{\color{cyan} \Large Lecture Notes} \chead{\LARGE Limits - Graphical Approach} \rhead{} \cfoot{} \lfoot{\small \copyright \; Hidegkuti, 2013} \rfoot{\small Last revised: January 14, 2014} \textwidth 7.5in \textheight 9.5in \setlength{\headheight}{27pt} \rhead{page \thepage} \setlength{\parindent}{0pt} \begin{document} \begin{enumerate} \item Determine the limits of the function based on its graph shown.\bigskip \bigskip\ %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion a) \ $\lim\limits_{x\rightarrow 3^{-}}f\left( x\right) $\bigskip b) \ $\lim\limits_{x\rightarrow 3^{+}}f\left( x\right) $\bigskip c) \ $\lim\limits_{x\rightarrow 3}f\left( x\right) $\bigskip d) \ $f\left( 3\right) $\bigskip e) \ $\lim\limits_{x\rightarrow 6^{-}}f\left( x\right) $\bigskip f) \ $\lim\limits_{x\rightarrow 6^{+}}f\left( x\right) $\bigskip g) \ $\lim\limits_{x\rightarrow 6}f\left( x\right) $\bigskip h) \ $f\left( 6\right) $\bigskip\ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \FRAME{dtbpF}{4.0145in}{3.2119in}{0pt}{}{}{pic1.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 4.0145in;height 3.2119in;depth 0pt;original-width 7.229in;original-height 5.7086in;cropleft "0";croptop "1";cropright "1.0106";cropbottom "0";filename 'pic1.bmp';file-properties "XNPEU";}} \item Determine each of the following based on the picture shown.\bigskip \bigskip\ %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion a) \ $\lim\limits_{x\rightarrow 1^{-}}f\left( x\right) $\bigskip b) \ $\lim\limits_{x\rightarrow 1^{+}}f\left( x\right) $\bigskip c) \ $\lim\limits_{x\rightarrow 1}f\left( x\right) $\bigskip d) \ $f\left( 1\right) $\bigskip e) \ $\lim\limits_{x\rightarrow 7^{-}}f\left( x\right) $\bigskip f) \ $\lim\limits_{x\rightarrow 7^{+}}f\left( x\right) $\bigskip g) \ $\lim\limits_{x\rightarrow 7}f\left( x\right) $\bigskip h) \ $f\left( 7\right) $ \bigskip\ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \FRAME{dtbpF}{4.0145in}{3.2119in}{0pt}{}{}{pic2.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 4.0145in;height 3.2119in;depth 0pt;original-width 7.6147in;original-height 5.156in;cropleft "0";croptop "0.9395";cropright "0.7989";cropbottom "0";filename 'pic2.bmp';file-properties "XNPEU";}} \pagebreak \end{enumerate} \begin{center} {\LARGE Answers\bigskip \bigskip } \end{center} \begin{enumerate} \item %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion a) \ $\lim\limits_{x\rightarrow 3^{-}}f\left( x\right) =5$\bigskip b) \ $\lim\limits_{x\rightarrow 3^{+}}f\left( x\right) =2$\bigskip c) \ $\lim\limits_{x\rightarrow 3}f\left( x\right) =\func{undefined}$\bigskip d) \ $f\left( 3\right) =\func{undefined}$\bigskip e) \ $\lim\limits_{x\rightarrow 6^{-}}f\left( x\right) =3$\bigskip f) \ $\lim\limits_{x\rightarrow 6^{+}}f\left( x\right) =3$\bigskip g) \ $\lim\limits_{x\rightarrow 6}f\left( x\right) =3$\bigskip h) \ $f\left( 6\right) =1$ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \vspace{1in} \item %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion a) \ $\lim\limits_{x\rightarrow 1^{-}}f\left( x\right) =2$\bigskip b) \ $\lim\limits_{x\rightarrow 1^{+}}f\left( x\right) =3$\bigskip c) \ $\lim\limits_{x\rightarrow 1}f\left( x\right) =\func{undefined}$\bigskip d) \ $f\left( 1\right) =2$\bigskip e) \ $\lim\limits_{x\rightarrow 7^{-}}f\left( x\right) =4$\bigskip f) \ $\lim\limits_{x\rightarrow 7^{+}}f\left( x\right) =4$\bigskip g) \ $\lim\limits_{x\rightarrow 7}f\left( x\right) =4$\bigskip h) \ $f\left( 7\right) =\func{undefined}$ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \end{enumerate} \vspace{1in} \vspace{2.5in} \bigskip {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}