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Find the oblique (slant) asymptote of \(\displaystyle f(x) = \frac{x^{2} - 13 x + 42}{x - 9}\).
The oblique asymptote is the whole part without the remainder after polynomial long division.
\begin{question}Find the oblique (slant) asymptote of $f(x) = \frac{x^{2} - 13 x + 42}{x - 9}$.
\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 - 13*x + 42}{x - 9} $};
\end{tikzpicture}
\end{center}
Dividing gives the quotient $x - 4$ and the remainder $6$. The oblique asymptote is $y = x - 4$. }
\end{question}
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\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} - 13 x + 42}{x - 9} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E%7B2%7D%20-%2013%20x%20%2B%2042%7D%7Bx%20-%209%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{x^{2} - 13 x + 42}{x - 9} " data-equation-content=" \displaystyle f(x) = \frac{x^{2} - 13 x + 42}{x - 9} " /> . </p> </p><p> <p>The oblique asymptote is the whole part without the remainder after polynomial long division. <br>
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Dividing gives the quotient <img class="equation_image" title=" \displaystyle x - 4 " src="/equation_images/%20%5Cdisplaystyle%20x%20-%204%20" alt="LaTeX: \displaystyle x - 4 " data-equation-content=" \displaystyle x - 4 " /> and the remainder <img class="equation_image" title=" \displaystyle 6 " src="/equation_images/%20%5Cdisplaystyle%206%20" alt="LaTeX: \displaystyle 6 " data-equation-content=" \displaystyle 6 " /> . The oblique asymptote is <img class="equation_image" title=" \displaystyle y = x - 4 " src="/equation_images/%20%5Cdisplaystyle%20y%20%3D%20x%20-%204%20" alt="LaTeX: \displaystyle y = x - 4 " data-equation-content=" \displaystyle y = x - 4 " /> . </p> </p>