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Questions: Algebra BusinessCalculus
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Evaluate the limit \(\displaystyle \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7}\)
The limit is an indeterminate form of the type \(\displaystyle \frac{\infty}{\infty}\). Using L'Hospitial's rule 3 times gives: \begin{equation*} \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} = \lim_{x \to -\infty}\frac{18 x^{2} - 14 x - 2}{27 x^{2} - 6 x - 8} = \lim_{x \to -\infty}\frac{2 \left(18 x - 7\right)}{6 \left(9 x - 1\right)} = \lim_{x \to -\infty}\frac{36}{54} = \frac{2}{3} \end{equation*}
\begin{question}Evaluate the limit $\lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7}$
\soln{9cm}{The limit is an indeterminate form of the type $\frac{\infty}{\infty}$. Using L'Hospitial's rule 3 times gives: \begin{equation*} \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} = \lim_{x \to -\infty}\frac{18 x^{2} - 14 x - 2}{27 x^{2} - 6 x - 8} = \lim_{x \to -\infty}\frac{2 \left(18 x - 7\right)}{6 \left(9 x - 1\right)} = \lim_{x \to -\infty}\frac{36}{54} = \frac{2}{3} \end{equation*}}
\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>Evaluate the limit <img class="equation_image" title=" \displaystyle \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cfrac%7B6%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20-%202%20x%20%2B%202%7D%7B9%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20-%208%20x%20%2B%207%7D%20" alt="LaTeX: \displaystyle \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} " data-equation-content=" \displaystyle \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} " /> </p> </p><p> <p>The limit is an indeterminate form of the type <img class="equation_image" title=" \displaystyle \frac{\infty}{\infty} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B%5Cinfty%7D%7B%5Cinfty%7D%20" alt="LaTeX: \displaystyle \frac{\infty}{\infty} " data-equation-content=" \displaystyle \frac{\infty}{\infty} " /> . Using L'Hospitial's rule 3 times gives: <img class="equation_image" title=" \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} = \lim_{x \to -\infty}\frac{18 x^{2} - 14 x - 2}{27 x^{2} - 6 x - 8} = \lim_{x \to -\infty}\frac{2 \left(18 x - 7\right)}{6 \left(9 x - 1\right)} = \lim_{x \to -\infty}\frac{36}{54} = \frac{2}{3} " src="/equation_images/%20%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cfrac%7B6%20x%5E%7B3%7D%20-%207%20x%5E%7B2%7D%20-%202%20x%20%2B%202%7D%7B9%20x%5E%7B3%7D%20-%203%20x%5E%7B2%7D%20-%208%20x%20%2B%207%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cfrac%7B18%20x%5E%7B2%7D%20-%2014%20x%20-%202%7D%7B27%20x%5E%7B2%7D%20-%206%20x%20-%208%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cfrac%7B2%20%5Cleft%2818%20x%20-%207%5Cright%29%7D%7B6%20%5Cleft%289%20x%20-%201%5Cright%29%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20-%5Cinfty%7D%5Cfrac%7B36%7D%7B54%7D%20%3D%20%5Cfrac%7B2%7D%7B3%7D%20%20" alt="LaTeX: \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} = \lim_{x \to -\infty}\frac{18 x^{2} - 14 x - 2}{27 x^{2} - 6 x - 8} = \lim_{x \to -\infty}\frac{2 \left(18 x - 7\right)}{6 \left(9 x - 1\right)} = \lim_{x \to -\infty}\frac{36}{54} = \frac{2}{3} " data-equation-content=" \lim_{x \to -\infty}\frac{6 x^{3} - 7 x^{2} - 2 x + 2}{9 x^{3} - 3 x^{2} - 8 x + 7} = \lim_{x \to -\infty}\frac{18 x^{2} - 14 x - 2}{27 x^{2} - 6 x - 8} = \lim_{x \to -\infty}\frac{2 \left(18 x - 7\right)}{6 \left(9 x - 1\right)} = \lim_{x \to -\infty}\frac{36}{54} = \frac{2}{3} " /> </p> </p>