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Calculus
Limits
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Evaluate the limit \(\displaystyle \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4}\)

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 -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} = \lim_{x \to -2}\frac{4 x - 6}{- 8 x - 6} = \frac{4 (-2) - 6}{- 8 (-2) - 6} = - \frac{7}{5} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Evaluate the limit $\lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4}$
    \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 -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} = \lim_{x \to -2}\frac{4 x - 6}{- 8 x - 6} = \frac{4 (-2) - 6}{- 8 (-2) - 6} = - \frac{7}{5} \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 -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-2%7D%5Cfrac%7B2%20x%5E%7B2%7D%20-%206%20x%20-%2020%7D%7B-%204%20x%5E%7B2%7D%20-%206%20x%20%2B%204%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} " data-equation-content=" \displaystyle \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} " /> </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 -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} = \lim_{x \to -2}\frac{4 x - 6}{- 8 x - 6} = \frac{4 (-2) - 6}{- 8 (-2) - 6} = - \frac{7}{5}  " src="/equation_images/%20%20%5Clim_%7Bx%20%5Cto%20-2%7D%5Cfrac%7B2%20x%5E%7B2%7D%20-%206%20x%20-%2020%7D%7B-%204%20x%5E%7B2%7D%20-%206%20x%20%2B%204%7D%20%3D%20%5Clim_%7Bx%20%5Cto%20-2%7D%5Cfrac%7B4%20x%20-%206%7D%7B-%208%20x%20-%206%7D%20%3D%20%5Cfrac%7B4%20%28-2%29%20-%206%7D%7B-%208%20%28-2%29%20-%206%7D%20%3D%20-%20%5Cfrac%7B7%7D%7B5%7D%20%20" alt="LaTeX:   \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} = \lim_{x \to -2}\frac{4 x - 6}{- 8 x - 6} = \frac{4 (-2) - 6}{- 8 (-2) - 6} = - \frac{7}{5}  " data-equation-content="  \lim_{x \to -2}\frac{2 x^{2} - 6 x - 20}{- 4 x^{2} - 6 x + 4} = \lim_{x \to -2}\frac{4 x - 6}{- 8 x - 6} = \frac{4 (-2) - 6}{- 8 (-2) - 6} = - \frac{7}{5}  " /> </p> </p>