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Find the vertical asymptote(s) and hole(s) of the rational function \(\displaystyle f(x)=\frac{ x^{2} + 16 x + 60 }{ x^{2} + 16 x + 60 }\)

Factoring gives \(\displaystyle f(x)=\frac{ \left(x + 6\right) \left(x + 10\right) }{ \left(x + 6\right) \left(x + 10\right) }\). The factor \(\displaystyle x + 6\) reduces out. The reduced function is \(\displaystyle f(x) = 1\). This gives a hole at \(\displaystyle \left(-6,1\right) \). The limit from the left is \(\displaystyle \lim_{x \to -10^-} 1 = 1\). The limit from the right is \(\displaystyle \lim_{x \to -10^+} 1 = 1\). This gives the vertical asymptote as \(\displaystyle x = -10\).

Download \(\LaTeX\) 

\begin{question}Find the vertical asymptote(s) and hole(s) of the rational function $f(x)=\frac{ x^{2} + 16 x + 60 }{ x^{2} + 16 x + 60 }$
    \soln{6cm}{Factoring gives $f(x)=\frac{ \left(x + 6\right) \left(x + 10\right) }{ \left(x + 6\right) \left(x + 10\right) }$. The factor $x + 6$ reduces out. The reduced function is $f(x) = 1$. This gives a hole at $\left(-6,1\right) $. The limit from the left is $\lim_{x \to -10^-} 1 = 1$. The limit from the right is $\lim_{x \to -10^+} 1 = 1$. This gives the vertical asymptote as $x = -10$.}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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    \soln{9cm}{The solution goes here.}

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HTML for Canvas 
<p> <p>Find the vertical asymptote(s) and hole(s) of the rational function  <img class="equation_image" title=" \displaystyle f(x)=\frac{ x^{2} + 16 x + 60 }{ x^{2} + 16 x + 60 } " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%5Cfrac%7B%20x%5E%7B2%7D%20%2B%2016%20x%20%2B%2060%20%7D%7B%20x%5E%7B2%7D%20%2B%2016%20x%20%2B%2060%20%7D%20" alt="LaTeX:  \displaystyle f(x)=\frac{ x^{2} + 16 x + 60 }{ x^{2} + 16 x + 60 } " data-equation-content=" \displaystyle f(x)=\frac{ x^{2} + 16 x + 60 }{ x^{2} + 16 x + 60 } " /> </p> </p>
HTML for Canvas
<p> <p>Factoring gives  <img class="equation_image" title=" \displaystyle f(x)=\frac{ \left(x + 6\right) \left(x + 10\right) }{ \left(x + 6\right) \left(x + 10\right) } " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%3D%5Cfrac%7B%20%5Cleft%28x%20%2B%206%5Cright%29%20%5Cleft%28x%20%2B%2010%5Cright%29%20%7D%7B%20%5Cleft%28x%20%2B%206%5Cright%29%20%5Cleft%28x%20%2B%2010%5Cright%29%20%7D%20" alt="LaTeX:  \displaystyle f(x)=\frac{ \left(x + 6\right) \left(x + 10\right) }{ \left(x + 6\right) \left(x + 10\right) } " data-equation-content=" \displaystyle f(x)=\frac{ \left(x + 6\right) \left(x + 10\right) }{ \left(x + 6\right) \left(x + 10\right) } " /> . The factor  <img class="equation_image" title=" \displaystyle x + 6 " src="/equation_images/%20%5Cdisplaystyle%20x%20%2B%206%20" alt="LaTeX:  \displaystyle x + 6 " data-equation-content=" \displaystyle x + 6 " />  reduces out. The reduced function is  <img class="equation_image" title=" \displaystyle f(x) = 1 " src="/equation_images/%20%5Cdisplaystyle%20f%28x%29%20%3D%201%20" alt="LaTeX:  \displaystyle f(x) = 1 " data-equation-content=" \displaystyle f(x) = 1 " /> . This gives a hole at  <img class="equation_image" title=" \displaystyle \left(-6,1\right)  " src="/equation_images/%20%5Cdisplaystyle%20%5Cleft%28-6%2C1%5Cright%29%20%20" alt="LaTeX:  \displaystyle \left(-6,1\right)  " data-equation-content=" \displaystyle \left(-6,1\right)  " /> . The limit from the left is  <img class="equation_image" title=" \displaystyle \lim_{x \to -10^-} 1 = 1 " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-10%5E-%7D%201%20%3D%201%20" alt="LaTeX:  \displaystyle \lim_{x \to -10^-} 1 = 1 " data-equation-content=" \displaystyle \lim_{x \to -10^-} 1 = 1 " /> . The limit from the right is  <img class="equation_image" title=" \displaystyle \lim_{x \to -10^+} 1 = 1 " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-10%5E%2B%7D%201%20%3D%201%20" alt="LaTeX:  \displaystyle \lim_{x \to -10^+} 1 = 1 " data-equation-content=" \displaystyle \lim_{x \to -10^+} 1 = 1 " /> . This gives the vertical asymptote as  <img class="equation_image" title=" \displaystyle x = -10 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-10%20" alt="LaTeX:  \displaystyle x = -10 " data-equation-content=" \displaystyle x = -10 " /> .</p> </p>