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Questions: Algebra BusinessCalculus
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Find the horizontal and vertical asymptote(s) of \(\displaystyle f(x) = \frac{x^{2} + 4 x - 96}{7 x^{2} + 21 x + 14}\), 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 \(\displaystyle y=\frac{1}{7}\). The vertical asymptotes are the zeros of the denominator. \(\displaystyle 7 x^{2} + 21 x + 14=0 \iff 7 \left(x + 1\right) \left(x + 2\right)=0\). The vertical asymptotes are \(\displaystyle x=-2\) and \(\displaystyle x=-1\).
\begin{question}Find the horizontal and vertical asymptote(s) of $f(x) = \frac{x^{2} + 4 x - 96}{7 x^{2} + 21 x + 14}$, 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 $y=\frac{1}{7}$. The vertical asymptotes are the zeros of the denominator. $7 x^{2} + 21 x + 14=0 \iff 7 \left(x + 1\right) \left(x + 2\right)=0$. The vertical asymptotes are $x=-2$ and $x=-1$.}
\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} + 4 x - 96}{7 x^{2} + 21 x + 14} " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%20%5Cfrac%7Bx%5E%7B2%7D%20%2B%204%20x%20-%2096%7D%7B7%20x%5E%7B2%7D%20%2B%2021%20x%20%2B%2014%7D%20" alt="LaTeX: \displaystyle f(x) = \frac{x^{2} + 4 x - 96}{7 x^{2} + 21 x + 14} " data-equation-content=" \displaystyle f(x) = \frac{x^{2} + 4 x - 96}{7 x^{2} + 21 x + 14} " /> , 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}{7} " src="/equation_images/%20%5Cdisplaystyle%20y%3D%5Cfrac%7B1%7D%7B7%7D%20" alt="LaTeX: \displaystyle y=\frac{1}{7} " data-equation-content=" \displaystyle y=\frac{1}{7} " /> . The vertical asymptotes are the zeros of the denominator. <img class="equation_image" title=" \displaystyle 7 x^{2} + 21 x + 14=0 \iff 7 \left(x + 1\right) \left(x + 2\right)=0 " src="/equation_images/%20%5Cdisplaystyle%207%20x%5E%7B2%7D%20%2B%2021%20x%20%2B%2014%3D0%20%5Ciff%207%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%3D0%20" alt="LaTeX: \displaystyle 7 x^{2} + 21 x + 14=0 \iff 7 \left(x + 1\right) \left(x + 2\right)=0 " data-equation-content=" \displaystyle 7 x^{2} + 21 x + 14=0 \iff 7 \left(x + 1\right) \left(x + 2\right)=0 " /> . The vertical asymptotes are <img class="equation_image" title=" \displaystyle x=-2 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-2%20" alt="LaTeX: \displaystyle x=-2 " data-equation-content=" \displaystyle x=-2 " /> and <img class="equation_image" title=" \displaystyle x=-1 " src="/equation_images/%20%5Cdisplaystyle%20x%3D-1%20" alt="LaTeX: \displaystyle x=-1 " data-equation-content=" \displaystyle x=-1 " /> .</p> </p>