\(\text{www.the}\beta\text{etafunction.com}\)
Home
Login
Questions: Algebra BusinessCalculus
Please login to create an exam or a quiz.
Find the oblique (slant) asymptote of \(\displaystyle f(x) = \frac{x^{2} - x - 56}{x - 5}\).
The oblique asymptote is the whole part without the remainder after polynomial long division.
\begin{question}Find the oblique (slant) asymptote of $f(x) = \frac{x^{2} - x - 56}{x - 5}$.
\soln{9cm}{The oblique asymptote is the whole part without the remainder after polynomial long division. \newline
\begin{center}\begin{tikzpicture}
\draw (0,0) node{$ \polylongdiv{x^2 - x - 56}{x - 5} $};
\end{tikzpicture}
\end{center}
Dividing gives the quotient $x + 4$ and the remainder $-36$. The oblique asymptote is $y = x + 4$. }
\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>Find the oblique (slant) asymptote of <img class="equation_image" title=" \displaystyle f(x) = \frac{x^{2} - x - 56}{x - 5} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E%7B2%7D%20-%20x%20-%2056%7D%7Bx%20-%205%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{x^{2} - x - 56}{x - 5} " data-equation-content=" \displaystyle f(x) = \frac{x^{2} - x - 56}{x - 5} " /> . </p> </p><p> <p>The oblique asymptote is the whole part without the remainder after polynomial long division. <br>
<center><?xml version="1.0" encoding="UTF-8"?>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="79.666pt" height="15.581pt" viewBox="0 0 79.666 15.581" 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 3.328125 -3.015625 C 3.390625 -3.265625 3.625 -4.1875 4.3125 -4.1875 C 4.359375 -4.1875 4.609375 -4.1875 4.8125 -4.0625 C 4.53125 -4 4.34375 -3.765625 4.34375 -3.515625 C 4.34375 -3.359375 4.453125 -3.171875 4.71875 -3.171875 C 4.9375 -3.171875 5.25 -3.34375 5.25 -3.75 C 5.25 -4.265625 4.671875 -4.40625 4.328125 -4.40625 C 3.75 -4.40625 3.40625 -3.875 3.28125 -3.65625 C 3.03125 -4.3125 2.5 -4.40625 2.203125 -4.40625 C 1.171875 -4.40625 0.59375 -3.125 0.59375 -2.875 C 0.59375 -2.765625 0.703125 -2.765625 0.71875 -2.765625 C 0.796875 -2.765625 0.828125 -2.796875 0.84375 -2.875 C 1.1875 -3.9375 1.84375 -4.1875 2.1875 -4.1875 C 2.375 -4.1875 2.71875 -4.09375 2.71875 -3.515625 C 2.71875 -3.203125 2.546875 -2.546875 2.1875 -1.140625 C 2.03125 -0.53125 1.671875 -0.109375 1.234375 -0.109375 C 1.171875 -0.109375 0.953125 -0.109375 0.734375 -0.234375 C 0.984375 -0.296875 1.203125 -0.5 1.203125 -0.78125 C 1.203125 -1.046875 0.984375 -1.125 0.84375 -1.125 C 0.53125 -1.125 0.296875 -0.875 0.296875 -0.546875 C 0.296875 -0.09375 0.78125 0.109375 1.21875 0.109375 C 1.890625 0.109375 2.25 -0.59375 2.265625 -0.640625 C 2.390625 -0.28125 2.75 0.109375 3.34375 0.109375 C 4.375 0.109375 4.9375 -1.171875 4.9375 -1.421875 C 4.9375 -1.53125 4.859375 -1.53125 4.828125 -1.53125 C 4.734375 -1.53125 4.71875 -1.484375 4.6875 -1.421875 C 4.359375 -0.34375 3.6875 -0.109375 3.375 -0.109375 C 2.984375 -0.109375 2.828125 -0.421875 2.828125 -0.765625 C 2.828125 -0.984375 2.875 -1.203125 2.984375 -1.640625 Z M 3.328125 -3.015625 "/>
</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 3.515625 -1.265625 L 3.28125 -1.265625 C 3.265625 -1.109375 3.1875 -0.703125 3.09375 -0.640625 C 3.046875 -0.59375 2.515625 -0.59375 2.40625 -0.59375 L 1.125 -0.59375 C 1.859375 -1.234375 2.109375 -1.4375 2.515625 -1.765625 C 3.03125 -2.171875 3.515625 -2.609375 3.515625 -3.265625 C 3.515625 -4.109375 2.78125 -4.625 1.890625 -4.625 C 1.03125 -4.625 0.4375 -4.015625 0.4375 -3.375 C 0.4375 -3.03125 0.734375 -2.984375 0.8125 -2.984375 C 0.96875 -2.984375 1.171875 -3.109375 1.171875 -3.359375 C 1.171875 -3.484375 1.125 -3.734375 0.765625 -3.734375 C 0.984375 -4.21875 1.453125 -4.375 1.78125 -4.375 C 2.484375 -4.375 2.84375 -3.828125 2.84375 -3.265625 C 2.84375 -2.65625 2.40625 -2.1875 2.1875 -1.9375 L 0.515625 -0.265625 C 0.4375 -0.203125 0.4375 -0.1875 0.4375 0 L 3.3125 0 Z M 3.515625 -1.265625 "/>
</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 6.5625 -2.296875 C 6.734375 -2.296875 6.921875 -2.296875 6.921875 -2.5 C 6.921875 -2.6875 6.734375 -2.6875 6.5625 -2.6875 L 1.171875 -2.6875 C 1 -2.6875 0.828125 -2.6875 0.828125 -2.5 C 0.828125 -2.296875 1 -2.296875 1.171875 -2.296875 Z M 6.5625 -2.296875 "/>
</symbol>
<symbol overflow="visible" id="glyph3-0">
<path style="stroke:none;" d=""/>
</symbol>
<symbol overflow="visible" id="glyph3-1">
<path style="stroke:none;" d="M 4.46875 -2 C 4.46875 -3.1875 3.65625 -4.1875 2.578125 -4.1875 C 2.109375 -4.1875 1.671875 -4.03125 1.3125 -3.671875 L 1.3125 -5.625 C 1.515625 -5.5625 1.84375 -5.5 2.15625 -5.5 C 3.390625 -5.5 4.09375 -6.40625 4.09375 -6.53125 C 4.09375 -6.59375 4.0625 -6.640625 3.984375 -6.640625 C 3.984375 -6.640625 3.953125 -6.640625 3.90625 -6.609375 C 3.703125 -6.515625 3.21875 -6.3125 2.546875 -6.3125 C 2.15625 -6.3125 1.6875 -6.390625 1.21875 -6.59375 C 1.140625 -6.625 1.125 -6.625 1.109375 -6.625 C 1 -6.625 1 -6.546875 1 -6.390625 L 1 -3.4375 C 1 -3.265625 1 -3.1875 1.140625 -3.1875 C 1.21875 -3.1875 1.234375 -3.203125 1.28125 -3.265625 C 1.390625 -3.421875 1.75 -3.96875 2.5625 -3.96875 C 3.078125 -3.96875 3.328125 -3.515625 3.40625 -3.328125 C 3.5625 -2.953125 3.59375 -2.578125 3.59375 -2.078125 C 3.59375 -1.71875 3.59375 -1.125 3.34375 -0.703125 C 3.109375 -0.3125 2.734375 -0.0625 2.28125 -0.0625 C 1.5625 -0.0625 0.984375 -0.59375 0.8125 -1.171875 C 0.84375 -1.171875 0.875 -1.15625 0.984375 -1.15625 C 1.3125 -1.15625 1.484375 -1.40625 1.484375 -1.640625 C 1.484375 -1.890625 1.3125 -2.140625 0.984375 -2.140625 C 0.84375 -2.140625 0.5 -2.0625 0.5 -1.609375 C 0.5 -0.75 1.1875 0.21875 2.296875 0.21875 C 3.453125 0.21875 4.46875 -0.734375 4.46875 -2 Z M 4.46875 -2 "/>
</symbol>
<symbol overflow="visible" id="glyph3-2">
<path style="stroke:none;" d="M 1.3125 -3.265625 L 1.3125 -3.515625 C 1.3125 -6.03125 2.546875 -6.390625 3.0625 -6.390625 C 3.296875 -6.390625 3.71875 -6.328125 3.9375 -5.984375 C 3.78125 -5.984375 3.390625 -5.984375 3.390625 -5.546875 C 3.390625 -5.234375 3.625 -5.078125 3.84375 -5.078125 C 4 -5.078125 4.3125 -5.171875 4.3125 -5.5625 C 4.3125 -6.15625 3.875 -6.640625 3.046875 -6.640625 C 1.765625 -6.640625 0.421875 -5.359375 0.421875 -3.15625 C 0.421875 -0.484375 1.578125 0.21875 2.5 0.21875 C 3.609375 0.21875 4.5625 -0.71875 4.5625 -2.03125 C 4.5625 -3.296875 3.671875 -4.25 2.5625 -4.25 C 1.890625 -4.25 1.515625 -3.75 1.3125 -3.265625 Z M 2.5 -0.0625 C 1.875 -0.0625 1.578125 -0.65625 1.515625 -0.8125 C 1.328125 -1.28125 1.328125 -2.078125 1.328125 -2.25 C 1.328125 -3.03125 1.65625 -4.03125 2.546875 -4.03125 C 2.71875 -4.03125 3.171875 -4.03125 3.484375 -3.40625 C 3.65625 -3.046875 3.65625 -2.53125 3.65625 -2.046875 C 3.65625 -1.5625 3.65625 -1.0625 3.484375 -0.703125 C 3.1875 -0.109375 2.734375 -0.0625 2.5 -0.0625 Z M 2.5 -0.0625 "/>
</symbol>
</g>
</defs>
<g id="surface1">
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="3.321" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph1-1" x="9.014" y="7.815"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="15.698" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="25.66" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="33.568" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph3-1" x="43.531" y="11.43"/>
<use xlink:href="#glyph3-2" x="48.5123" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph0-1" x="53.493" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph2-1" x="61.401" y="11.43"/>
</g>
<g style="fill:rgb(0%,0%,0%);fill-opacity:1;">
<use xlink:href="#glyph3-1" x="71.364" y="11.43"/>
</g>
</g>
</svg>
</center>
Dividing gives the quotient <img class="equation_image" title=" \displaystyle x + 4 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%204%20" alt="LaTeX: \displaystyle x + 4 " data-equation-content=" \displaystyle x + 4 " /> and the remainder <img class="equation_image" title=" \displaystyle -36 " src="/equation_images/%20%5Cdisplaystyle%20-36%20" alt="LaTeX: \displaystyle -36 " data-equation-content=" \displaystyle -36 " /> . The oblique asymptote is <img class="equation_image" title=" \displaystyle y = x + 4 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20x%20%2B%204%20" alt="LaTeX: \displaystyle y = x + 4 " data-equation-content=" \displaystyle y = x + 4 " /> . </p> </p>