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Building the fraction by muliplying by the conjuate gives: \(\displaystyle \lim_{x\to-\infty}\frac{12 x + \sqrt{144 x^{2} + 12 x + 1}}{1}\cdot\left(\frac{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}\right)\) Simplifying gives \(\displaystyle \lim_{x \to -\infty} \frac{12 x + 1}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}\)factoring an \(\displaystyle x^2\) out of the radical in the denomiator and using the fact that \(\displaystyle \sqrt{x^2}=|x|\) gives \(\displaystyle \lim_{x \to -\infty} \frac{x \left(12 + \frac{1}{x}\right)}{- 12 x + \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} \left|{x}\right|}\) Using the fact that \(\displaystyle |x| =-x\) when \(\displaystyle x<0\) and reducing gives \(\displaystyle \lim_{x \to -\infty}\left(\frac{12 + \frac{1}{x}}{- \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} - 12}\right) = - \frac{1}{2}\)

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\begin{question}Evaluate the limit $\displaystyle\lim_{x \to -\infty}\left(12 x + \sqrt{144 x^{2} + 12 x + 1}\right)$
    \soln{9cm}{Building the fraction by muliplying by the conjuate gives: $\lim_{x\to-\infty}\frac{12 x + \sqrt{144 x^{2} + 12 x + 1}}{1}\cdot\left(\frac{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}\right)$ Simplifying gives $\lim_{x \to -\infty} \frac{12 x + 1}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}$factoring an $x^2$ out of the radical in the denomiator and using the fact that $\sqrt{x^2}=|x|$ gives $\lim_{x \to -\infty} \frac{x \left(12 + \frac{1}{x}\right)}{- 12 x + \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} \left|{x}\right|}$ Using the fact that $|x| =-x$ when $x<0$ and reducing gives $\lim_{x \to -\infty}\left(\frac{12 + \frac{1}{x}}{- \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} - 12}\right) = - \frac{1}{2}$ }

\end{question}

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

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HTML for Canvas

<p> <p>Evaluate the limit  <img class="equation_image" title=" \displaystyle \displaystyle\lim_{x \to -\infty}\left(12 x + \sqrt{144 x^{2} + 12 x + 1}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Cdisplaystyle%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cleft%2812%20x%20%2B%20%5Csqrt%7B144%20x%5E%7B2%7D%20%2B%2012%20x%20%2B%201%7D%5Cright%29%20" alt="LaTeX:  \displaystyle \displaystyle\lim_{x \to -\infty}\left(12 x + \sqrt{144 x^{2} + 12 x + 1}\right) " data-equation-content=" \displaystyle \displaystyle\lim_{x \to -\infty}\left(12 x + \sqrt{144 x^{2} + 12 x + 1}\right) " /> </p> </p>

HTML for Canvas

<p> <p>Building the fraction by muliplying by the conjuate gives:  <img class="equation_image" title=" \displaystyle \lim_{x\to-\infty}\frac{12 x + \sqrt{144 x^{2} + 12 x + 1}}{1}\cdot\left(\frac{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}\right) " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%5Cto-%5Cinfty%7D%5Cfrac%7B12%20x%20%2B%20%5Csqrt%7B144%20x%5E%7B2%7D%20%2B%2012%20x%20%2B%201%7D%7D%7B1%7D%5Ccdot%5Cleft%28%5Cfrac%7B-%2012%20x%20%2B%20%5Csqrt%7B144%20x%5E%7B2%7D%20%2B%2012%20x%20%2B%201%7D%7D%7B-%2012%20x%20%2B%20%5Csqrt%7B144%20x%5E%7B2%7D%20%2B%2012%20x%20%2B%201%7D%7D%5Cright%29%20" alt="LaTeX:  \displaystyle \lim_{x\to-\infty}\frac{12 x + \sqrt{144 x^{2} + 12 x + 1}}{1}\cdot\left(\frac{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}\right) " data-equation-content=" \displaystyle \lim_{x\to-\infty}\frac{12 x + \sqrt{144 x^{2} + 12 x + 1}}{1}\cdot\left(\frac{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}}\right) " />  Simplifying gives  <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty} \frac{12 x + 1}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7B12%20x%20%2B%201%7D%7B-%2012%20x%20%2B%20%5Csqrt%7B144%20x%5E%7B2%7D%20%2B%2012%20x%20%2B%201%7D%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -\infty} \frac{12 x + 1}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}} " data-equation-content=" \displaystyle \lim_{x \to -\infty} \frac{12 x + 1}{- 12 x + \sqrt{144 x^{2} + 12 x + 1}} " /> factoring an  <img class="equation_image" title=" \displaystyle x^2 " src="/equation_images/%20%5Cdisplaystyle%20x%5E2%20" alt="LaTeX:  \displaystyle x^2 " data-equation-content=" \displaystyle x^2 " />  out of the radical in the denomiator and using the fact that  <img class="equation_image" title=" \displaystyle \sqrt{x^2}=|x| " src="/equation_images/%20%5Cdisplaystyle%20%5Csqrt%7Bx%5E2%7D%3D%7Cx%7C%20" alt="LaTeX:  \displaystyle \sqrt{x^2}=|x| " data-equation-content=" \displaystyle \sqrt{x^2}=|x| " />  gives  <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty} \frac{x \left(12 + \frac{1}{x}\right)}{- 12 x + \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} \left|{x}\right|} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%20%5Cfrac%7Bx%20%5Cleft%2812%20%2B%20%5Cfrac%7B1%7D%7Bx%7D%5Cright%29%7D%7B-%2012%20x%20%2B%20%5Csqrt%7B144%20%2B%20%5Cfrac%7B12%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%7D%20%5Cleft%7C%7Bx%7D%5Cright%7C%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -\infty} \frac{x \left(12 + \frac{1}{x}\right)}{- 12 x + \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} \left|{x}\right|} " data-equation-content=" \displaystyle \lim_{x \to -\infty} \frac{x \left(12 + \frac{1}{x}\right)}{- 12 x + \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} \left|{x}\right|} " />  Using the fact that  <img class="equation_image" title=" \displaystyle |x| =-x " src="/equation_images/%20%5Cdisplaystyle%20%7Cx%7C%20%3D-x%20" alt="LaTeX:  \displaystyle |x| =-x " data-equation-content=" \displaystyle |x| =-x " />  when  <img class="equation_image" title=" \displaystyle x<0 " src="/equation_images/%20%5Cdisplaystyle%20x%3C0%20" alt="LaTeX:  \displaystyle x<0 " data-equation-content=" \displaystyle x<0 " />  and reducing gives  <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty}\left(\frac{12 + \frac{1}{x}}{- \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} - 12}\right) = - \frac{1}{2} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cleft%28%5Cfrac%7B12%20%2B%20%5Cfrac%7B1%7D%7Bx%7D%7D%7B-%20%5Csqrt%7B144%20%2B%20%5Cfrac%7B12%7D%7Bx%7D%20%2B%20%5Cfrac%7B1%7D%7Bx%5E%7B2%7D%7D%7D%20-%2012%7D%5Cright%29%20%3D%20-%20%5Cfrac%7B1%7D%7B2%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -\infty}\left(\frac{12 + \frac{1}{x}}{- \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} - 12}\right) = - \frac{1}{2} " data-equation-content=" \displaystyle \lim_{x \to -\infty}\left(\frac{12 + \frac{1}{x}}{- \sqrt{144 + \frac{12}{x} + \frac{1}{x^{2}}} - 12}\right) = - \frac{1}{2} " />  </p> </p>