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Questions: Algebra BusinessCalculus
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Find the horizontal and vertical asymptote(s) of \(\displaystyle f(x) = \frac{x^{2} - 19 x + 88}{9 x^{2} - 18 x - 135}\), if they exist.
The degree of the numerator is the same as the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients \(\displaystyle y=\frac{1}{9}\). The vertical asymptotes are the zeros of the denominator. \(\displaystyle 9 x^{2} - 18 x - 135=0 \iff 9 \left(x - 5\right) \left(x + 3\right)=0\). The vertical asymptotes are \(\displaystyle x=-3\) and \(\displaystyle x=5\).
\begin{question}Find the horizontal and vertical asymptote(s) of $f(x) = \frac{x^{2} - 19 x + 88}{9 x^{2} - 18 x - 135}$, if they exist.
\soln{9cm}{The degree of the numerator is the same as the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients $y=\frac{1}{9}$. The vertical asymptotes are the zeros of the denominator. $9 x^{2} - 18 x - 135=0 \iff 9 \left(x - 5\right) \left(x + 3\right)=0$. The vertical asymptotes are $x=-3$ and $x=5$.}
\end{question}
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\soln{9cm}{The solution goes here.}
\end{question}\end{document}<p> <p>Find the horizontal and vertical asymptote(s) of <img class="equation_image" title=" \displaystyle f(x) = \frac{x^{2} - 19 x + 88}{9 x^{2} - 18 x - 135} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E%7B2%7D%20-%2019%20x%20%2B%2088%7D%7B9%20x%5E%7B2%7D%20-%2018%20x%20-%20135%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{x^{2} - 19 x + 88}{9 x^{2} - 18 x - 135} " data-equation-content=" \displaystyle f(x) = \frac{x^{2} - 19 x + 88}{9 x^{2} - 18 x - 135} " /> , if they exist.</p> </p><p> <p>The degree of the numerator is the same as the degree of the denominator, so the horizontal asymptote is the ratio of the leading coefficients <img class="equation_image" title=" \displaystyle y=\frac{1}{9} " src="/equation_images/%20%5Cdisplaystyle%20y%3D%5Cfrac%7B1%7D%7B9%7D%20" alt="LaTeX: \displaystyle y=\frac{1}{9} " data-equation-content=" \displaystyle y=\frac{1}{9} " /> . The vertical asymptotes are the zeros of the denominator. <img class="equation_image" title=" \displaystyle 9 x^{2} - 18 x - 135=0 \iff 9 \left(x - 5\right) \left(x + 3\right)=0 " src="/equation_images/%20%5Cdisplaystyle%209%20x%5E%7B2%7D%20-%2018%20x%20-%20135%3D0%20%5Ciff%209%20%5Cleft%28x%20-%205%5Cright%29%20%5Cleft%28x%20%2B%203%5Cright%29%3D0%20" alt="LaTeX: \displaystyle 9 x^{2} - 18 x - 135=0 \iff 9 \left(x - 5\right) \left(x + 3\right)=0 " data-equation-content=" \displaystyle 9 x^{2} - 18 x - 135=0 \iff 9 \left(x - 5\right) \left(x + 3\right)=0 " /> . The vertical asymptotes are <img class="equation_image" title=" \displaystyle x=-3 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-3%20" alt="LaTeX: \displaystyle x=-3 " data-equation-content=" \displaystyle x=-3 " /> and <img class="equation_image" title=" \displaystyle x=5 " src="/equation_images/%20%5Cdisplaystyle%20x%3D5%20" alt="LaTeX: \displaystyle x=5 " data-equation-content=" \displaystyle x=5 " /> .</p> </p>