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Calculus
Limits
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Find the limit, if it exists \(\displaystyle \lim_{x \to 4 }\frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 }\)

Building the fraction by multiplying by the LCD in the numerator and denominator gives: \begin{equation*} \frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 }\left(\frac{ 4 x }{ 4 x } \right)=\frac{ 20 - 5 x }{ 4 x \left(x - 4\right) } = \frac{ - 5 \left(x - 4\right) }{ 4 x \left(x - 4\right) } = - \frac{5}{4 x}\end{equation*}The reduced function is continuous at \(\displaystyle x = 4\) and by the evaluation theorem \(\displaystyle \lim_{x \to 4 }- \frac{5}{4 x} = - \frac{5}{16}\)

Download \(\LaTeX\) 

\begin{question}Find the limit, if it exists $\lim_{x \to 4 }\frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 }$ 
    \soln{9cm}{Building the fraction by multiplying by the LCD in the numerator and denominator gives: \begin{equation*} \frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 }\left(\frac{ 4 x }{ 4 x } \right)=\frac{ 20 - 5 x }{ 4 x \left(x - 4\right) } = \frac{ - 5 \left(x - 4\right) }{ 4 x \left(x - 4\right) } = - \frac{5}{4 x}\end{equation*}The reduced function is continuous at $x = 4$ and by the evaluation theorem $\lim_{x \to 4 }- \frac{5}{4 x} = - \frac{5}{16}$}

\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 4 }\frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 } " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%204%20%7D%5Cfrac%7B%20%5Cfrac%7B5%7D%7Bx%7D%20-%20%5Cfrac%7B5%7D%7B4%7D%20%7D%7B%20x%20-%204%20%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to 4 }\frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 } " data-equation-content=" \displaystyle \lim_{x \to 4 }\frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 } " />  </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{5}{x} - \frac{5}{4} }{ x - 4 }\left(\frac{ 4 x }{ 4 x } \right)=\frac{ 20 - 5 x }{ 4 x \left(x - 4\right) } = \frac{ - 5 \left(x - 4\right) }{ 4 x \left(x - 4\right) } = - \frac{5}{4 x} " src="/equation_images/%20%20%5Cfrac%7B%20%5Cfrac%7B5%7D%7Bx%7D%20-%20%5Cfrac%7B5%7D%7B4%7D%20%7D%7B%20x%20-%204%20%7D%5Cleft%28%5Cfrac%7B%204%20x%20%7D%7B%204%20x%20%7D%20%5Cright%29%3D%5Cfrac%7B%2020%20-%205%20x%20%7D%7B%204%20x%20%5Cleft%28x%20-%204%5Cright%29%20%7D%20%3D%20%5Cfrac%7B%20-%205%20%5Cleft%28x%20-%204%5Cright%29%20%7D%7B%204%20x%20%5Cleft%28x%20-%204%5Cright%29%20%7D%20%3D%20-%20%5Cfrac%7B5%7D%7B4%20x%7D%20" alt="LaTeX:   \frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 }\left(\frac{ 4 x }{ 4 x } \right)=\frac{ 20 - 5 x }{ 4 x \left(x - 4\right) } = \frac{ - 5 \left(x - 4\right) }{ 4 x \left(x - 4\right) } = - \frac{5}{4 x} " data-equation-content="  \frac{ \frac{5}{x} - \frac{5}{4} }{ x - 4 }\left(\frac{ 4 x }{ 4 x } \right)=\frac{ 20 - 5 x }{ 4 x \left(x - 4\right) } = \frac{ - 5 \left(x - 4\right) }{ 4 x \left(x - 4\right) } = - \frac{5}{4 x} " /> The reduced function is continuous at  <img class="equation_image" title=" \displaystyle x = 4 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%204%20" alt="LaTeX:  \displaystyle x = 4 " data-equation-content=" \displaystyle x = 4 " />  and by the evaluation theorem  <img class="equation_image" title=" \displaystyle \lim_{x \to 4 }- \frac{5}{4 x} = - \frac{5}{16} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%204%20%7D-%20%5Cfrac%7B5%7D%7B4%20x%7D%20%3D%20-%20%5Cfrac%7B5%7D%7B16%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to 4 }- \frac{5}{4 x} = - \frac{5}{16} " data-equation-content=" \displaystyle \lim_{x \to 4 }- \frac{5}{4 x} = - \frac{5}{16} " /> </p> </p>