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Calculus
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Find the limit, if it exists \(\displaystyle \lim_{x \to -2 }\frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }\)

Building the fraction by multiplying by the LCD in the numerator and denominator gives: \begin{equation*} \frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }\left(\frac{ 4 x^{2} }{ 4 x^{2} } \right)=\frac{ 28 - 7 x^{2} }{ 4 x^{2} \left(x + 2\right) } = \frac{ - 7 \left(x - 2\right) \left(x + 2\right) }{ 4 x^{2} \left(x + 2\right) } = - \frac{7 \left(x - 2\right)}{4 x^{2}}\end{equation*}The reduced function is continuous at \(\displaystyle x = -2\) and by the evaluation theorem \(\displaystyle \lim_{x \to -2 }- \frac{7 \left(x - 2\right)}{4 x^{2}} = \frac{7}{4}\)

Download \(\LaTeX\) 

\begin{question}Find the limit, if it exists $\lim_{x \to -2 }\frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }$ 
    \soln{9cm}{Building the fraction by multiplying by the LCD in the numerator and denominator gives: \begin{equation*} \frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }\left(\frac{ 4 x^{2} }{ 4 x^{2} } \right)=\frac{ 28 - 7 x^{2} }{ 4 x^{2} \left(x + 2\right) } = \frac{ - 7 \left(x - 2\right) \left(x + 2\right) }{ 4 x^{2} \left(x + 2\right) } = - \frac{7 \left(x - 2\right)}{4 x^{2}}\end{equation*}The reduced function is continuous at $x = -2$ and by the evaluation theorem $\lim_{x \to -2 }- \frac{7 \left(x - 2\right)}{4 x^{2}} = \frac{7}{4}$}

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Find the limit, if it exists  <img class="equation_image" title=" \displaystyle \lim_{x \to -2 }\frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 } " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-2%20%7D%5Cfrac%7B%20%5Cfrac%7B7%7D%7Bx%5E%7B2%7D%7D%20-%20%5Cfrac%7B7%7D%7B4%7D%20%7D%7B%20x%20%2B%202%20%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -2 }\frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 } " data-equation-content=" \displaystyle \lim_{x \to -2 }\frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 } " />  </p> </p>
HTML for Canvas
<p> <p>Building the fraction by multiplying by the LCD in the numerator and denominator gives:  <img class="equation_image" title="  \frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }\left(\frac{ 4 x^{2} }{ 4 x^{2} } \right)=\frac{ 28 - 7 x^{2} }{ 4 x^{2} \left(x + 2\right) } = \frac{ - 7 \left(x - 2\right) \left(x + 2\right) }{ 4 x^{2} \left(x + 2\right) } = - \frac{7 \left(x - 2\right)}{4 x^{2}} " src="/equation_images/%20%20%5Cfrac%7B%20%5Cfrac%7B7%7D%7Bx%5E%7B2%7D%7D%20-%20%5Cfrac%7B7%7D%7B4%7D%20%7D%7B%20x%20%2B%202%20%7D%5Cleft%28%5Cfrac%7B%204%20x%5E%7B2%7D%20%7D%7B%204%20x%5E%7B2%7D%20%7D%20%5Cright%29%3D%5Cfrac%7B%2028%20-%207%20x%5E%7B2%7D%20%7D%7B%204%20x%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%20%7D%20%3D%20%5Cfrac%7B%20-%207%20%5Cleft%28x%20-%202%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%20%7D%7B%204%20x%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%20%7D%20%3D%20-%20%5Cfrac%7B7%20%5Cleft%28x%20-%202%5Cright%29%7D%7B4%20x%5E%7B2%7D%7D%20" alt="LaTeX:   \frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }\left(\frac{ 4 x^{2} }{ 4 x^{2} } \right)=\frac{ 28 - 7 x^{2} }{ 4 x^{2} \left(x + 2\right) } = \frac{ - 7 \left(x - 2\right) \left(x + 2\right) }{ 4 x^{2} \left(x + 2\right) } = - \frac{7 \left(x - 2\right)}{4 x^{2}} " data-equation-content="  \frac{ \frac{7}{x^{2}} - \frac{7}{4} }{ x + 2 }\left(\frac{ 4 x^{2} }{ 4 x^{2} } \right)=\frac{ 28 - 7 x^{2} }{ 4 x^{2} \left(x + 2\right) } = \frac{ - 7 \left(x - 2\right) \left(x + 2\right) }{ 4 x^{2} \left(x + 2\right) } = - \frac{7 \left(x - 2\right)}{4 x^{2}} " /> The reduced function is continuous at  <img class="equation_image" title=" \displaystyle x = -2 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-2%20" alt="LaTeX:  \displaystyle x = -2 " data-equation-content=" \displaystyle x = -2 " />  and by the evaluation theorem  <img class="equation_image" title=" \displaystyle \lim_{x \to -2 }- \frac{7 \left(x - 2\right)}{4 x^{2}} = \frac{7}{4} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-2%20%7D-%20%5Cfrac%7B7%20%5Cleft%28x%20-%202%5Cright%29%7D%7B4%20x%5E%7B2%7D%7D%20%3D%20%5Cfrac%7B7%7D%7B4%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -2 }- \frac{7 \left(x - 2\right)}{4 x^{2}} = \frac{7}{4} " data-equation-content=" \displaystyle \lim_{x \to -2 }- \frac{7 \left(x - 2\right)}{4 x^{2}} = \frac{7}{4} " /> </p> </p>