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Calculus
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Evaluate the limit \(\displaystyle \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3}\)

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 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} = \lim_{x \to 3}\frac{10 x + 10}{23 - 16 x} = \frac{10 (3) + 10}{23 - 16 (3)} = - \frac{8}{5} \end{equation*}

Download \(\LaTeX\) 

\begin{question}Evaluate the limit $\lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3}$
    \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 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} = \lim_{x \to 3}\frac{10 x + 10}{23 - 16 x} = \frac{10 (3) + 10}{23 - 16 (3)} = - \frac{8}{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 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%203%7D%5Cfrac%7B5%20x%5E%7B2%7D%20%2B%2010%20x%20-%2075%7D%7B-%208%20x%5E%7B2%7D%20%2B%2023%20x%20%2B%203%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} " data-equation-content=" \displaystyle \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} " /> </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 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} = \lim_{x \to 3}\frac{10 x + 10}{23 - 16 x} = \frac{10 (3) + 10}{23 - 16 (3)} = - \frac{8}{5}  " src="/equation_images/%20%20%5Clim_%7Bx%20%5Cto%203%7D%5Cfrac%7B5%20x%5E%7B2%7D%20%2B%2010%20x%20-%2075%7D%7B-%208%20x%5E%7B2%7D%20%2B%2023%20x%20%2B%203%7D%20%3D%20%5Clim_%7Bx%20%5Cto%203%7D%5Cfrac%7B10%20x%20%2B%2010%7D%7B23%20-%2016%20x%7D%20%3D%20%5Cfrac%7B10%20%283%29%20%2B%2010%7D%7B23%20-%2016%20%283%29%7D%20%3D%20-%20%5Cfrac%7B8%7D%7B5%7D%20%20" alt="LaTeX:   \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} = \lim_{x \to 3}\frac{10 x + 10}{23 - 16 x} = \frac{10 (3) + 10}{23 - 16 (3)} = - \frac{8}{5}  " data-equation-content="  \lim_{x \to 3}\frac{5 x^{2} + 10 x - 75}{- 8 x^{2} + 23 x + 3} = \lim_{x \to 3}\frac{10 x + 10}{23 - 16 x} = \frac{10 (3) + 10}{23 - 16 (3)} = - \frac{8}{5}  " /> </p> </p>