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Questions: Algebra BusinessCalculus
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Evaluate the limit
The limit is an indeterminate form of the type
\begin{question}Evaluate the limit $\lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1}$
\soln{9cm}{The limit is an indeterminate form of the type $\frac{\infty}{\infty}$. Using L'Hospitial's rule 2 times gives: \begin{equation*} \lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} = \lim_{x \to \infty}\frac{4 x + 8}{18 x - 2} = \lim_{x \to \infty}\frac{4}{18} = \frac{2}{9} \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{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%5Cfrac%7B2%20x%5E%7B2%7D%20%2B%208%20x%20-%209%7D%7B9%20x%5E%7B2%7D%20-%202%20x%20%2B%201%7D%20" alt="LaTeX: \displaystyle \lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} " data-equation-content=" \displaystyle \lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} " /> </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 2 times gives: <img class="equation_image" title=" \lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} = \lim_{x \to \infty}\frac{4 x + 8}{18 x - 2} = \lim_{x \to \infty}\frac{4}{18} = \frac{2}{9} " src="/equation_images/%20%20%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%5Cfrac%7B2%20x%5E%7B2%7D%20%2B%208%20x%20-%209%7D%7B9%20x%5E%7B2%7D%20-%202%20x%20%2B%201%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%5Cfrac%7B4%20x%20%2B%208%7D%7B18%20x%20-%202%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20%5Cinfty%7D%5Cfrac%7B4%7D%7B18%7D%20%3D%20%5Cfrac%7B2%7D%7B9%7D%20%20" alt="LaTeX: \lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} = \lim_{x \to \infty}\frac{4 x + 8}{18 x - 2} = \lim_{x \to \infty}\frac{4}{18} = \frac{2}{9} " data-equation-content=" \lim_{x \to \infty}\frac{2 x^{2} + 8 x - 9}{9 x^{2} - 2 x + 1} = \lim_{x \to \infty}\frac{4 x + 8}{18 x - 2} = \lim_{x \to \infty}\frac{4}{18} = \frac{2}{9} " /> </p> </p>