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Calculus
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Evaluate the limit \(\displaystyle \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30}\)

The limit is an indeterminate form of the type \(\displaystyle \frac{0}{0}\). Using L'Hospitial's rule and then the evaluation theorem gives: \begin{equation*} \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} = \lim_{x \to -6}\frac{4 x - 4}{18 x + 49} = \frac{4 (-6) - 4}{18 (-6) + 49} = \frac{28}{59} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Evaluate the limit $\lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30}$
    \soln{9cm}{The limit is an indeterminate form of the type $\frac{0}{0}$. Using L'Hospitial's rule and then the evaluation theorem gives: \begin{equation*} \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} = \lim_{x \to -6}\frac{4 x - 4}{18 x + 49} = \frac{4 (-6) - 4}{18 (-6) + 49} = \frac{28}{59} \end{equation*}}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Evaluate the limit  <img class="equation_image" title=" \displaystyle \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-6%7D%5Cfrac%7B2%20x%5E%7B2%7D%20-%204%20x%20-%2096%7D%7B9%20x%5E%7B2%7D%20%2B%2049%20x%20-%2030%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} " data-equation-content=" \displaystyle \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} " /> </p> </p>
HTML for Canvas
<p> <p>The limit is an indeterminate form of the type  <img class="equation_image" title=" \displaystyle \frac{0}{0} " src="/equation_images/%20%5Cdisplaystyle%20%5Cfrac%7B0%7D%7B0%7D%20" alt="LaTeX:  \displaystyle \frac{0}{0} " data-equation-content=" \displaystyle \frac{0}{0} " /> . Using L'Hospitial's rule and then the evaluation theorem gives:  <img class="equation_image" title="  \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} = \lim_{x \to -6}\frac{4 x - 4}{18 x + 49} = \frac{4 (-6) - 4}{18 (-6) + 49} = \frac{28}{59}  " src="/equation_images/%20%20%5Clim_%7Bx%20%5Cto%20-6%7D%5Cfrac%7B2%20x%5E%7B2%7D%20-%204%20x%20-%2096%7D%7B9%20x%5E%7B2%7D%20%2B%2049%20x%20-%2030%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20-6%7D%5Cfrac%7B4%20x%20-%204%7D%7B18%20x%20%2B%2049%7D%20%3D%20%5Cfrac%7B4%20%28-6%29%20-%204%7D%7B18%20%28-6%29%20%2B%2049%7D%20%3D%20%5Cfrac%7B28%7D%7B59%7D%20%20" alt="LaTeX:   \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} = \lim_{x \to -6}\frac{4 x - 4}{18 x + 49} = \frac{4 (-6) - 4}{18 (-6) + 49} = \frac{28}{59}  " data-equation-content="  \lim_{x \to -6}\frac{2 x^{2} - 4 x - 96}{9 x^{2} + 49 x - 30} = \lim_{x \to -6}\frac{4 x - 4}{18 x + 49} = \frac{4 (-6) - 4}{18 (-6) + 49} = \frac{28}{59}  " /> </p> </p>