\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Use the graph to evaluate the the following:
\begin{tabular}{|c|c|}\hline
\(\displaystyle \displaystyle\lim_{x\to -2^-}f(x)\)
\(\displaystyle =-2\) & \(\displaystyle \displaystyle\lim_{x\to 4^-}f(x)\)
\(\displaystyle =-1\) \\ \hline
\(\displaystyle \displaystyle\lim_{x\to -2^+}f(x)\)
\(\displaystyle =-1\) & \(\displaystyle \displaystyle\lim_{x\to 4^+}f(x)\)
\(\displaystyle =-1\) \\ \hline
\(\displaystyle \displaystyle\lim_{x\to -2}f(x)\)
\(\displaystyle =\text{Does Not Exist}\) & \(\displaystyle \displaystyle\lim_{x\to 4}f(x)\)
\(\displaystyle =-1\) \\ \hline
\(\displaystyle f(-2)\)\(\displaystyle =-2\) & \(\displaystyle f(4)\)\(\displaystyle =-8\) \\ \hline
\end{tabular}
\begin{tikzpicture}[font=\LARGE, scale=.6]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
\foreach \x in { -10,-8,...,10 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -10,-8,...,10 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[latex-,very thick] (-10,-2) -- (-2,-2);
\draw[very thick] (-2, -1) parabola bend (1, 4.6) (4, -1);
\draw[-latex, very thick] (4,-1) -- (10,1);
\draw[fill=white] (-2,-1) circle (4pt);
\draw[fill=white] (4,-1) circle (4pt);
\draw[fill=black] (-2,-2) circle (4pt);
\draw[fill=black] (4,-8) circle (4pt);
\end{tikzpicture}
\begin{question}Use the graph to evaluate the the following:
\begin{tabular}{p{10cm}p{10cm}}
$\displaystyle\lim_{x\to -2^-}f(x)$
& $\displaystyle\lim_{x\to 4^-}f(x)$
\\
$\displaystyle\lim_{x\to -2^+}f(x)$
& $\displaystyle\lim_{x\to 4^+}f(x)$
\\
$\displaystyle\lim_{x\to -2}f(x)$
& $\displaystyle\lim_{x\to 4}f(x)$
\\
$f(-2)$ & $f(4)$ \\
\end{tabular}\newline
\begin{tikzpicture}[font=\LARGE, scale=.6]
\draw[very thick, color=blue, opacity=0.3, xstep=1, ystep=1] (-10,-10) grid (10,10);
\draw[latex-latex, very thick] (-10.4,0)--(10.4,0) node[right]{$x$};
\draw[latex-latex, very thick] (0,-10.4)--(0,10.4) node[above]{$y$};
\foreach \x in { -10,-8,...,10 }{
\pgfmathsetmacro{\xtick}{\x != 0 ? "\x" : ""}
\draw (\x, .2) -- (\x,-.2) node[below]{$\xtick$};
}
\foreach \y in { -10,-8,...,10 }{
\pgfmathsetmacro{\ytick}{\y != 0 ? "\y" : ""}
\draw (.2, \y) -- (-.2, \y) node[left]{$\ytick$};
}
\draw[latex-,very thick] (-10,-2) -- (-2,-2);
\draw[very thick] (-2, -1) parabola bend (1, 4.6) (4, -1);
\draw[-latex, very thick] (4,-1) -- (10,1);
\draw[fill=white] (-2,-1) circle (4pt);
\draw[fill=white] (4,-1) circle (4pt);
\draw[fill=black] (-2,-2) circle (4pt);
\draw[fill=black] (4,-8) circle (4pt);
\end{tikzpicture}
\soln{9cm}{\begin{tabular}{|c|c|}\hline
$\displaystyle\lim_{x\to -2^-}f(x)$
$=-2$ & $\displaystyle\lim_{x\to 4^-}f(x)$
$=-1$ \\ \hline
$\displaystyle\lim_{x\to -2^+}f(x)$
$=-1$ & $\displaystyle\lim_{x\to 4^+}f(x)$
$=-1$ \\ \hline
$\displaystyle\lim_{x\to -2}f(x)$
$=\text{Does Not Exist}$ & $\displaystyle\lim_{x\to 4}f(x)$
$=-1$ \\ \hline
$f(-2)$$=-2$ & $f(4)$$=-8$ \\ \hline
\end{tabular}\newline
}
\end{question}
\documentclass{article}
\usepackage{tikz}
\usepackage{amsmath}
\usepackage[margin=2cm]{geometry}
\usepackage{tcolorbox}
\newcounter{ExamNumber}
\newcounter{questioncount}
\stepcounter{questioncount}
\newenvironment{question}{{\noindent\bfseries Question \arabic{questioncount}.}}{\stepcounter{questioncount}}
\renewcommand{\labelenumi}{{\bfseries (\alph{enumi})}}
\newif\ifShowSolution
\newcommand{\soln}[2]{%
\ifShowSolution%
\noindent\begin{tcolorbox}[colframe=blue,title=Solution]#2\end{tcolorbox}\else%
\vspace{#1}%
\fi%
}%
\newcommand{\hideifShowSolution}[1]{%
\ifShowSolution%
%
\else%
#1%
\fi%
}%
\everymath{\displaystyle}
\ShowSolutiontrue
\begin{document}\begin{question}(10pts) The question goes here!
\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Use the graph to evaluate the the following:
<?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="381.692pt" height="374.549pt" viewBox="0 0 381.692 374.549" version="1.1">
<defs>
<g>
<symbol overflow="visible" id="glyph0-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph0-1">
<path style="stroke:none;" d="M 8.15625 -7.03125 C 7.609375 -6.921875 7.40625 -6.515625 7.40625 -6.1875 C 7.40625 -5.765625 7.734375 -5.625 7.96875 -5.625 C 8.484375 -5.625 8.84375 -6.078125 8.84375 -6.546875 C 8.84375 -7.265625 8.03125 -7.59375 7.296875 -7.59375 C 6.25 -7.59375 5.671875 -6.5625 5.515625 -6.234375 C 5.109375 -7.53125 4.046875 -7.59375 3.734375 -7.59375 C 1.984375 -7.59375 1.046875 -5.34375 1.046875 -4.953125 C 1.046875 -4.890625 1.125 -4.796875 1.234375 -4.796875 C 1.375 -4.796875 1.40625 -4.90625 1.453125 -4.96875 C 2.03125 -6.890625 3.1875 -7.25 3.6875 -7.25 C 4.453125 -7.25 4.609375 -6.53125 4.609375 -6.109375 C 4.609375 -5.734375 4.515625 -5.34375 4.3125 -4.515625 L 3.71875 -2.15625 C 3.46875 -1.125 2.96875 -0.171875 2.046875 -0.171875 C 1.96875 -0.171875 1.53125 -0.171875 1.171875 -0.390625 C 1.796875 -0.515625 1.921875 -1.03125 1.921875 -1.234375 C 1.921875 -1.578125 1.671875 -1.796875 1.34375 -1.796875 C 0.9375 -1.796875 0.484375 -1.421875 0.484375 -0.875 C 0.484375 -0.15625 1.296875 0.171875 2.03125 0.171875 C 2.859375 0.171875 3.4375 -0.484375 3.8125 -1.1875 C 4.078125 -0.171875 4.9375 0.171875 5.578125 0.171875 C 7.328125 0.171875 8.265625 -2.078125 8.265625 -2.46875 C 8.265625 -2.546875 8.203125 -2.625 8.09375 -2.625 C 7.9375 -2.625 7.921875 -2.53125 7.875 -2.390625 C 7.40625 -0.875 6.40625 -0.171875 5.625 -0.171875 C 5.03125 -0.171875 4.703125 -0.625 4.703125 -1.328125 C 4.703125 -1.703125 4.765625 -1.984375 5.046875 -3.109375 L 5.640625 -5.453125 C 5.90625 -6.484375 6.484375 -7.25 7.28125 -7.25 C 7.3125 -7.25 7.796875 -7.25 8.15625 -7.03125 Z M 8.15625 -7.03125 "/>
</symbol>
<symbol overflow="visible" id="glyph0-2">
<path style="stroke:none;" d="M 4.53125 1.921875 C 4.0625 2.578125 3.390625 3.171875 2.546875 3.171875 C 2.34375 3.171875 1.515625 3.140625 1.25 2.34375 C 1.3125 2.359375 1.390625 2.359375 1.421875 2.359375 C 1.953125 2.359375 2.296875 1.90625 2.296875 1.515625 C 2.296875 1.125 1.96875 0.984375 1.703125 0.984375 C 1.421875 0.984375 0.828125 1.1875 0.828125 2.03125 C 0.828125 2.90625 1.5625 3.515625 2.546875 3.515625 C 4.265625 3.515625 6.015625 1.921875 6.484375 0.015625 L 8.171875 -6.703125 C 8.203125 -6.78125 8.234375 -6.890625 8.234375 -6.984375 C 8.234375 -7.25 8.03125 -7.421875 7.765625 -7.421875 C 7.609375 -7.421875 7.25 -7.359375 7.109375 -6.828125 L 5.84375 -1.78125 C 5.75 -1.46875 5.75 -1.421875 5.609375 -1.234375 C 5.265625 -0.75 4.703125 -0.171875 3.875 -0.171875 C 2.90625 -0.171875 2.828125 -1.125 2.828125 -1.578125 C 2.828125 -2.5625 3.28125 -3.890625 3.75 -5.125 C 3.9375 -5.625 4.046875 -5.875 4.046875 -6.21875 C 4.046875 -6.9375 3.53125 -7.59375 2.6875 -7.59375 C 1.109375 -7.59375 0.46875 -5.09375 0.46875 -4.953125 C 0.46875 -4.890625 0.53125 -4.796875 0.65625 -4.796875 C 0.8125 -4.796875 0.828125 -4.875 0.890625 -5.109375 C 1.3125 -6.5625 1.96875 -7.25 2.640625 -7.25 C 2.796875 -7.25 3.078125 -7.25 3.078125 -6.6875 C 3.078125 -6.234375 2.890625 -5.734375 2.640625 -5.078125 C 1.796875 -2.828125 1.796875 -2.25 1.796875 -1.84375 C 1.796875 -0.203125 2.96875 0.171875 3.828125 0.171875 C 4.328125 0.171875 4.9375 0.015625 5.546875 -0.625 L 5.5625 -0.609375 C 5.296875 0.40625 5.125 1.078125 4.53125 1.921875 Z M 4.53125 1.921875 "/>
</symbol>
<symbol overflow="visible" id="glyph1-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph1-1">
<path style="stroke:none;" d="M 11.34375 -3.953125 C 11.640625 -3.953125 11.953125 -3.953125 11.953125 -4.3125 C 11.953125 -4.65625 11.640625 -4.65625 11.34375 -4.65625 L 2.03125 -4.65625 C 1.734375 -4.65625 1.421875 -4.65625 1.421875 -4.3125 C 1.421875 -3.953125 1.734375 -3.953125 2.03125 -3.953125 Z M 11.34375 -3.953125 "/>
</symbol>
<symbol overflow="visible" id="glyph2-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph2-1">
<path style="stroke:none;" d="M 4.578125 -11 C 4.578125 -11.359375 4.5625 -11.375 4.328125 -11.375 C 3.65625 -10.578125 2.640625 -10.3125 1.671875 -10.28125 C 1.625 -10.28125 1.53125 -10.28125 1.515625 -10.25 C 1.5 -10.203125 1.5 -10.171875 1.5 -9.8125 C 2.03125 -9.8125 2.921875 -9.921875 3.609375 -10.328125 L 3.609375 -1.25 C 3.609375 -0.65625 3.578125 -0.453125 2.09375 -0.453125 L 1.578125 -0.453125 L 1.578125 0 C 2.40625 -0.015625 3.265625 -0.03125 4.09375 -0.03125 C 4.921875 -0.03125 5.78125 -0.015625 6.609375 0 L 6.609375 -0.453125 L 6.09375 -0.453125 C 4.609375 -0.453125 4.578125 -0.640625 4.578125 -1.25 Z M 4.578125 -11 "/>
</symbol>
<symbol overflow="visible" id="glyph2-2">
<path style="stroke:none;" d="M 7.234375 -5.53125 C 7.234375 -6.578125 7.21875 -8.375 6.484375 -9.75 C 5.859375 -10.953125 4.84375 -11.375 3.9375 -11.375 C 3.109375 -11.375 2.0625 -11 1.40625 -9.765625 C 0.71875 -8.46875 0.65625 -6.875 0.65625 -5.53125 C 0.65625 -4.546875 0.671875 -3.046875 1.203125 -1.734375 C 1.953125 0.03125 3.265625 0.28125 3.9375 0.28125 C 4.734375 0.28125 5.9375 -0.046875 6.640625 -1.6875 C 7.15625 -2.875 7.234375 -4.265625 7.234375 -5.53125 Z M 3.9375 0 C 2.84375 0 2.1875 -0.953125 1.953125 -2.25 C 1.75 -3.265625 1.75 -4.75 1.75 -5.71875 C 1.75 -7.046875 1.75 -8.140625 1.984375 -9.1875 C 2.3125 -10.65625 3.265625 -11.109375 3.9375 -11.109375 C 4.65625 -11.109375 5.5625 -10.640625 5.890625 -9.234375 C 6.109375 -8.25 6.125 -7.09375 6.125 -5.71875 C 6.125 -4.59375 6.125 -3.21875 5.921875 -2.203125 C 5.5625 -0.328125 4.546875 0 3.9375 0 Z M 3.9375 0 "/>
</symbol>
<symbol overflow="visible" id="glyph2-3">
<path style="stroke:none;" d="M 4.6875 -6.203125 C 5.765625 -6.75 6.875 -7.578125 6.875 -8.90625 C 6.875 -10.46875 5.359375 -11.375 3.953125 -11.375 C 2.390625 -11.375 1.015625 -10.25 1.015625 -8.671875 C 1.015625 -8.25 1.125 -7.5 1.796875 -6.859375 C 1.96875 -6.6875 2.6875 -6.171875 3.15625 -5.84375 C 2.375 -5.4375 0.5625 -4.5 0.5625 -2.59375 C 0.5625 -0.828125 2.25 0.28125 3.921875 0.28125 C 5.765625 0.28125 7.3125 -1.046875 7.3125 -2.8125 C 7.3125 -4.375 6.265625 -5.09375 5.578125 -5.5625 Z M 2.421875 -7.71875 C 2.296875 -7.796875 1.59375 -8.328125 1.59375 -9.140625 C 1.59375 -10.1875 2.6875 -10.984375 3.921875 -10.984375 C 5.28125 -10.984375 6.28125 -10.015625 6.28125 -8.90625 C 6.28125 -7.296875 4.5 -6.390625 4.40625 -6.390625 C 4.390625 -6.390625 4.375 -6.390625 4.234375 -6.484375 Z M 5.59375 -4.1875 C 5.859375 -4 6.6875 -3.421875 6.6875 -2.375 C 6.6875 -1.109375 5.40625 -0.140625 3.953125 -0.140625 C 2.390625 -0.140625 1.203125 -1.25 1.203125 -2.625 C 1.203125 -3.984375 2.25 -5.109375 3.4375 -5.640625 Z M 5.59375 -4.1875 "/>
</symbol>
<symbol overflow="visible" id="glyph2-4">
<path style="stroke:none;" d="M 1.828125 -5.9375 C 1.828125 -10.0625 3.75 -10.984375 4.875 -10.984375 C 5.234375 -10.984375 6.109375 -10.921875 6.453125 -10.25 C 6.1875 -10.25 5.671875 -10.25 5.671875 -9.640625 C 5.671875 -9.171875 6.046875 -9.015625 6.28125 -9.015625 C 6.4375 -9.015625 6.90625 -9.09375 6.90625 -9.671875 C 6.90625 -10.765625 6.046875 -11.375 4.859375 -11.375 C 2.8125 -11.375 0.65625 -9.25 0.65625 -5.4375 C 0.65625 -0.75 2.59375 0.28125 3.984375 0.28125 C 5.640625 0.28125 7.234375 -1.21875 7.234375 -3.53125 C 7.234375 -5.6875 5.84375 -7.25 4.078125 -7.25 C 3.03125 -7.25 2.25 -6.5625 1.828125 -5.359375 Z M 3.984375 -0.140625 C 1.859375 -0.140625 1.859375 -3.3125 1.859375 -3.9375 C 1.859375 -5.1875 2.4375 -6.96875 4.046875 -6.96875 C 4.34375 -6.96875 5.1875 -6.96875 5.75 -5.78125 C 6.0625 -5.109375 6.0625 -4.40625 6.0625 -3.546875 C 6.0625 -2.625 6.0625 -1.921875 5.703125 -1.234375 C 5.328125 -0.53125 4.765625 -0.140625 3.984375 -0.140625 Z M 3.984375 -0.140625 "/>
</symbol>
<symbol overflow="visible" id="glyph2-5">
<path style="stroke:none;" d="M 5.78125 -11.140625 C 5.78125 -11.5 5.765625 -11.515625 5.453125 -11.515625 L 0.34375 -3.375 L 0.34375 -2.921875 L 4.78125 -2.921875 L 4.78125 -1.234375 C 4.78125 -0.625 4.75 -0.453125 3.546875 -0.453125 L 3.21875 -0.453125 L 3.21875 0 C 3.765625 -0.03125 4.703125 -0.03125 5.28125 -0.03125 C 5.875 -0.03125 6.796875 -0.03125 7.359375 0 L 7.359375 -0.453125 L 7.03125 -0.453125 C 5.8125 -0.453125 5.78125 -0.625 5.78125 -1.234375 L 5.78125 -2.921875 L 7.546875 -2.921875 L 7.546875 -3.375 L 5.78125 -3.375 Z M 4.84375 -10 L 4.84375 -3.375 L 0.6875 -3.375 Z M 4.84375 -10 "/>
</symbol>
<symbol overflow="visible" id="glyph2-6">
<path style="stroke:none;" d="M 7.1875 -2.671875 L 6.875 -2.671875 C 6.703125 -1.453125 6.5625 -1.234375 6.484375 -1.140625 C 6.40625 -1 5.171875 -1 4.921875 -1 L 1.625 -1 C 2.234375 -1.671875 3.4375 -2.890625 4.90625 -4.3125 C 5.953125 -5.296875 7.1875 -6.46875 7.1875 -8.171875 C 7.1875 -10.203125 5.5625 -11.375 3.75 -11.375 C 1.859375 -11.375 0.703125 -9.71875 0.703125 -8.15625 C 0.703125 -7.484375 1.203125 -7.40625 1.40625 -7.40625 C 1.578125 -7.40625 2.09375 -7.5 2.09375 -8.109375 C 2.09375 -8.640625 1.65625 -8.796875 1.40625 -8.796875 C 1.3125 -8.796875 1.203125 -8.78125 1.140625 -8.75 C 1.46875 -10.203125 2.46875 -10.9375 3.515625 -10.9375 C 5.015625 -10.9375 5.984375 -9.75 5.984375 -8.171875 C 5.984375 -6.6875 5.109375 -5.390625 4.125 -4.265625 L 0.703125 -0.390625 L 0.703125 0 L 6.765625 0 Z M 7.1875 -2.671875 "/>
</symbol>
</g>
</defs>
<g id="surface1">
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,100%);stroke-opacity:0.3;stroke-miterlimit:10;" d="M -170.080844 -170.082844 L 170.083219 -170.082844 M -170.080844 -153.075031 L 170.083219 -153.075031 M -170.080844 -136.067219 L 170.083219 -136.067219 M -170.080844 -119.059406 L 170.083219 -119.059406 M -170.080844 -102.047687 L 170.083219 -102.047687 M -170.080844 -85.039875 L 170.083219 -85.039875 M -170.080844 -68.032062 L 170.083219 -68.032062 M -170.080844 -51.02425 L 170.083219 -51.02425 M -170.080844 -34.016437 L 170.083219 -34.016437 M -170.080844 -17.008625 L 170.083219 -17.008625 M -170.080844 -0.0008125 L 170.083219 -0.0008125 M -170.080844 17.007 L 170.083219 17.007 M -170.080844 34.014813 L 170.083219 34.014813 M -170.080844 51.026531 L 170.083219 51.026531 M -170.080844 68.034344 L 170.083219 68.034344 M -170.080844 85.042156 L 170.083219 85.042156 M -170.080844 102.049969 L 170.083219 102.049969 M -170.080844 119.057781 L 170.083219 119.057781 M -170.080844 136.065594 L 170.083219 136.065594 M -170.080844 153.073406 L 170.083219 153.073406 M -170.080844 170.077313 L 170.083219 170.077313 M -170.080844 -170.082844 L -170.080844 170.081219 M -153.073031 -170.082844 L -153.073031 170.081219 M -136.065219 -170.082844 L -136.065219 170.081219 M -119.057406 -170.082844 L -119.057406 170.081219 M -102.049594 -170.082844 L -102.049594 170.081219 M -85.041781 -170.082844 L -85.041781 170.081219 M -68.033969 -170.082844 L -68.033969 170.081219 M -51.026156 -170.082844 L -51.026156 170.081219 M -34.018344 -170.082844 L -34.018344 170.081219 M -17.006625 -170.082844 L -17.006625 170.081219 M 0.0011875 -170.082844 L 0.0011875 170.081219 M 17.009 -170.082844 L 17.009 170.081219 M 34.016812 -170.082844 L 34.016812 170.081219 M 51.024625 -170.082844 L 51.024625 170.081219 M 68.032437 -170.082844 L 68.032437 170.081219 M 85.04025 -170.082844 L 85.04025 170.081219 M 102.048062 -170.082844 L 102.048062 170.081219 M 119.055875 -170.082844 L 119.055875 170.081219 M 136.067594 -170.082844 L 136.067594 170.081219 M 153.075406 -170.082844 L 153.075406 170.081219 M 170.075406 -170.082844 L 170.075406 170.081219 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -171.14725 -0.0008125 L 171.145719 -0.0008125 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 11.105469 194.882812 C 12.804688 195.199219 15.570312 196.15625 17.480469 197.273438 L 17.480469 192.492188 C 15.570312 193.605469 12.804688 194.5625 11.105469 194.882812 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 364.875 194.882812 C 363.175781 194.5625 360.414062 193.605469 358.5 192.492188 L 358.5 197.273438 C 360.414062 196.15625 363.175781 195.199219 364.875 194.882812 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="368.793" y="198.588"/>
</g>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 0.0011875 -171.145344 L 0.0011875 171.147625 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 187.992188 371.769531 C 188.308594 370.066406 189.265625 367.304688 190.382812 365.390625 L 185.601562 365.390625 C 186.714844 367.304688 187.671875 370.066406 187.992188 371.769531 "/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 187.992188 17.996094 C 187.671875 19.695312 186.714844 22.460938 185.601562 24.371094 L 190.382812 24.371094 C 189.265625 22.460938 188.308594 19.695312 187.992188 17.996094 "/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-2" x="183.573" y="10.733"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -170.080844 3.401531 L -170.080844 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="3.321" y="212.901"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="16.71" y="212.901"/>
<use xlink:href="#glyph2-2" x="24.604982" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -136.065219 3.401531 L -136.065219 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="41.284" y="212.901"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="54.674" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -102.049594 3.401531 L -102.049594 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="75.3" y="212.901"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="88.69" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -68.033969 3.401531 L -68.033969 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="109.316" y="212.901"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-5" x="122.706" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -34.018344 3.401531 L -34.018344 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="143.332" y="212.901"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-6" x="156.722" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 0.0011875 3.401531 L 0.0011875 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 34.016812 3.401531 L 34.016812 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-6" x="218.059" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 68.032437 3.401531 L 68.032437 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-5" x="252.076" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 102.048062 3.401531 L 102.048062 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="286.092" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 136.067594 3.401531 L 136.067594 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="320.108" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 170.083219 3.401531 L 170.083219 -3.403156 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="350.176" y="212.901"/>
<use xlink:href="#glyph2-2" x="358.070982" y="212.901"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 -170.082844 L -3.401156 -170.082844 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="151.889" y="369.794"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="165.279" y="369.794"/>
<use xlink:href="#glyph2-2" x="173.173982" y="369.794"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 -136.067219 L -3.401156 -136.067219 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="159.785" y="335.778"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="173.174" y="335.778"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 -102.047687 L -3.401156 -102.047687 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="159.785" y="301.762"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="173.174" y="301.762"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 -68.032062 L -3.401156 -68.032062 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="159.785" y="267.746"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-5" x="173.174" y="267.746"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 -34.016437 L -3.401156 -34.016437 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="159.785" y="233.73"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-6" x="173.175" y="233.73"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 -0.0008125 L -3.401156 -0.0008125 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 34.014813 L -3.401156 34.014813 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-6" x="173.174" y="166.415"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 68.034344 L -3.401156 68.034344 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-5" x="173.174" y="132.399"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 102.049969 L -3.401156 102.049969 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-4" x="173.174" y="98.383"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 136.065594 L -3.401156 136.065594 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-3" x="173.174" y="64.366"/>
</g>
<path style="fill:none;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 3.403531 170.081219 L -3.401156 170.081219 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="165.279" y="30.35"/>
<use xlink:href="#glyph2-2" x="173.173982" y="30.35"/>
</g>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -164.342563 -34.016437 L -34.018344 -34.016437 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 17.910156 228.898438 C 19.609375 229.21875 22.371094 230.171875 24.285156 231.289062 L 24.285156 226.507812 C 22.371094 227.625 19.609375 228.578125 17.910156 228.898438 "/>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -34.018344 -17.008625 C -28.276156 4.421063 -8.502719 78.237469 17.009 78.237469 C 42.520719 78.237469 62.294156 4.421063 68.032437 -17.008625 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill:none;stroke-width:1.19553;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 68.032437 -17.008625 L 164.637906 15.1945 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style=" stroke:none;fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;" d="M 358.074219 177.875 C 356.359375 178.109375 353.4375 178.074219 351.269531 177.621094 L 352.78125 182.160156 C 354.242188 180.496094 356.5625 178.714844 358.074219 177.875 "/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -31.623813 -17.008625 C -31.623813 -15.688312 -32.694125 -14.618 -34.018344 -14.618 C -35.338656 -14.618 -36.408969 -15.688312 -36.408969 -17.008625 C -36.408969 -18.328937 -35.338656 -19.39925 -34.018344 -19.39925 C -32.694125 -19.39925 -31.623813 -18.328937 -31.623813 -17.008625 Z M -31.623813 -17.008625 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 70.423062 -17.008625 C 70.423062 -15.688312 69.35275 -14.618 68.032437 -14.618 C 66.712125 -14.618 65.641812 -15.688312 65.641812 -17.008625 C 65.641812 -18.328937 66.712125 -19.39925 68.032437 -19.39925 C 69.35275 -19.39925 70.423062 -18.328937 70.423062 -17.008625 Z M 70.423062 -17.008625 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -31.623813 -34.016437 C -31.623813 -32.696125 -32.694125 -31.625812 -34.018344 -31.625812 C -35.338656 -31.625812 -36.408969 -32.696125 -36.408969 -34.016437 C -36.408969 -35.33675 -35.338656 -36.407062 -34.018344 -36.407062 C -32.694125 -36.407062 -31.623813 -35.33675 -31.623813 -34.016437 Z M -31.623813 -34.016437 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
<path style="fill-rule:nonzero;fill:rgb(0%,0%,0%);fill-opacity:1;stroke-width:0.3985;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 70.423062 -136.067219 C 70.423062 -134.746906 69.35275 -133.676594 68.032437 -133.676594 C 66.712125 -133.676594 65.641812 -134.746906 65.641812 -136.067219 C 65.641812 -137.387531 66.712125 -138.457844 68.032437 -138.457844 C 69.35275 -138.457844 70.423062 -137.387531 70.423062 -136.067219 Z M 70.423062 -136.067219 " transform="matrix(1,0,0,-1,187.991,194.882)"/>
</g>
</svg>
</p> </p>
<p> <p>\begin{tabular}{|c|c|}\hline
<img class="equation_image" title=" \displaystyle \displaystyle\lim_{x\to -2^-}f(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%5Cto%20-2%5E-%7Df%28x%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x\to -2^-}f(x) " data-equation-content=" \displaystyle \displaystyle\lim_{x\to -2^-}f(x) " />
<img class="equation_image" title=" \displaystyle =-2 " src="/equation_images/%20%5Cdisplaystyle%20%3D-2%20" alt="LaTeX: \displaystyle =-2 " data-equation-content=" \displaystyle =-2 " /> & <img class="equation_image" title=" \displaystyle \displaystyle\lim_{x\to 4^-}f(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%5Cto%204%5E-%7Df%28x%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x\to 4^-}f(x) " data-equation-content=" \displaystyle \displaystyle\lim_{x\to 4^-}f(x) " />
<img class="equation_image" title=" \displaystyle =-1 " src="/equation_images/%20%5Cdisplaystyle%20%3D-1%20" alt="LaTeX: \displaystyle =-1 " data-equation-content=" \displaystyle =-1 " /> \\ \hline
<img class="equation_image" title=" \displaystyle \displaystyle\lim_{x\to -2^+}f(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%5Cto%20-2%5E%2B%7Df%28x%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x\to -2^+}f(x) " data-equation-content=" \displaystyle \displaystyle\lim_{x\to -2^+}f(x) " />
<img class="equation_image" title=" \displaystyle =-1 " src="/equation_images/%20%5Cdisplaystyle%20%3D-1%20" alt="LaTeX: \displaystyle =-1 " data-equation-content=" \displaystyle =-1 " /> & <img class="equation_image" title=" \displaystyle \displaystyle\lim_{x\to 4^+}f(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%5Cto%204%5E%2B%7Df%28x%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x\to 4^+}f(x) " data-equation-content=" \displaystyle \displaystyle\lim_{x\to 4^+}f(x) " />
<img class="equation_image" title=" \displaystyle =-1 " src="/equation_images/%20%5Cdisplaystyle%20%3D-1%20" alt="LaTeX: \displaystyle =-1 " data-equation-content=" \displaystyle =-1 " /> \\ \hline
<img class="equation_image" title=" \displaystyle \displaystyle\lim_{x\to -2}f(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%5Cto%20-2%7Df%28x%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x\to -2}f(x) " data-equation-content=" \displaystyle \displaystyle\lim_{x\to -2}f(x) " />
<img class="equation_image" title=" \displaystyle =\text{Does Not Exist} " src="/equation_images/%20%5Cdisplaystyle%20%3D%5Ctext%7BDoes%20Not%20Exist%7D%20" alt="LaTeX: \displaystyle =\text{Does Not Exist} " data-equation-content=" \displaystyle =\text{Does Not Exist} " /> & <img class="equation_image" title=" \displaystyle \displaystyle\lim_{x\to 4}f(x) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%5Cto%204%7Df%28x%29%20" alt="LaTeX: \displaystyle \displaystyle\lim_{x\to 4}f(x) " data-equation-content=" \displaystyle \displaystyle\lim_{x\to 4}f(x) " />
<img class="equation_image" title=" \displaystyle =-1 " src="/equation_images/%20%5Cdisplaystyle%20%3D-1%20" alt="LaTeX: \displaystyle =-1 " data-equation-content=" \displaystyle =-1 " /> \\ \hline
<img class="equation_image" title=" \displaystyle f(-2) " src="/equation_images/%20%5Cdisplaystyle%20f%28-2%29%20" alt="LaTeX: \displaystyle f(-2) " data-equation-content=" \displaystyle f(-2) " /> <img class="equation_image" title=" \displaystyle =-2 " src="/equation_images/%20%5Cdisplaystyle%20%3D-2%20" alt="LaTeX: \displaystyle =-2 " data-equation-content=" \displaystyle =-2 " /> & <img class="equation_image" title=" \displaystyle f(4) " src="/equation_images/%20%5Cdisplaystyle%20f%284%29%20" alt="LaTeX: \displaystyle f(4) " data-equation-content=" \displaystyle f(4) " /> <img class="equation_image" title=" \displaystyle =-8 " src="/equation_images/%20%5Cdisplaystyle%20%3D-8%20" alt="LaTeX: \displaystyle =-8 " data-equation-content=" \displaystyle =-8 " /> \\ \hline
\end{tabular}<br>
</p> </p>