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Calculus
Limits
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Find the limit, if it exists \(\displaystyle \lim_{x \to -4 }\frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }\)

Building the fraction by multiplying by the conjugate in the numerator and denominator gives: \begin{equation*} \frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }\left(\frac{ \sqrt{x + 6} + \sqrt{2} }{ \sqrt{x + 6} + \sqrt{2} } \right)=\frac{ x + 4 }{ \left(x + 4\right) \left(\sqrt{x + 6} + \sqrt{2}\right) } = \frac{1}{\sqrt{x + 6} + \sqrt{2}} \end{equation*}The reduced function is continuous at \(\displaystyle x = -4\) and by the evaluation theorem \(\displaystyle \lim_{x \to -4 }\frac{1}{\sqrt{x + 6} + \sqrt{2}} = \frac{\sqrt{2}}{4}\)

Download \(\LaTeX\) 

\begin{question}Find the limit, if it exists $\lim_{x \to -4 }\frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }$ 
    \soln{9cm}{Building the fraction by multiplying by the conjugate in the numerator and denominator gives: \begin{equation*} \frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }\left(\frac{ \sqrt{x + 6} + \sqrt{2} }{ \sqrt{x + 6} + \sqrt{2} } \right)=\frac{ x + 4 }{ \left(x + 4\right) \left(\sqrt{x + 6} + \sqrt{2}\right) } = \frac{1}{\sqrt{x + 6} + \sqrt{2}} \end{equation*}The reduced function is continuous at $x = -4$ and by the evaluation theorem $\lim_{x \to -4 }\frac{1}{\sqrt{x + 6} + \sqrt{2}} = \frac{\sqrt{2}}{4}$}

\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 -4 }\frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 } " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-4%20%7D%5Cfrac%7B%20%5Csqrt%7Bx%20%2B%206%7D%20-%20%5Csqrt%7B2%7D%20%7D%7B%20x%20%2B%204%20%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -4 }\frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 } " data-equation-content=" \displaystyle \lim_{x \to -4 }\frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 } " />  </p> </p>
HTML for Canvas
<p> <p>Building the fraction by multiplying by the conjugate in the numerator and denominator gives:  <img class="equation_image" title="  \frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }\left(\frac{ \sqrt{x + 6} + \sqrt{2} }{ \sqrt{x + 6} + \sqrt{2} } \right)=\frac{ x + 4 }{ \left(x + 4\right) \left(\sqrt{x + 6} + \sqrt{2}\right) } = \frac{1}{\sqrt{x + 6} + \sqrt{2}}  " src="/equation_images/%20%20%5Cfrac%7B%20%5Csqrt%7Bx%20%2B%206%7D%20-%20%5Csqrt%7B2%7D%20%7D%7B%20x%20%2B%204%20%7D%5Cleft%28%5Cfrac%7B%20%5Csqrt%7Bx%20%2B%206%7D%20%2B%20%5Csqrt%7B2%7D%20%7D%7B%20%5Csqrt%7Bx%20%2B%206%7D%20%2B%20%5Csqrt%7B2%7D%20%7D%20%5Cright%29%3D%5Cfrac%7B%20x%20%2B%204%20%7D%7B%20%5Cleft%28x%20%2B%204%5Cright%29%20%5Cleft%28%5Csqrt%7Bx%20%2B%206%7D%20%2B%20%5Csqrt%7B2%7D%5Cright%29%20%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%20%2B%206%7D%20%2B%20%5Csqrt%7B2%7D%7D%20%20" alt="LaTeX:   \frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }\left(\frac{ \sqrt{x + 6} + \sqrt{2} }{ \sqrt{x + 6} + \sqrt{2} } \right)=\frac{ x + 4 }{ \left(x + 4\right) \left(\sqrt{x + 6} + \sqrt{2}\right) } = \frac{1}{\sqrt{x + 6} + \sqrt{2}}  " data-equation-content="  \frac{ \sqrt{x + 6} - \sqrt{2} }{ x + 4 }\left(\frac{ \sqrt{x + 6} + \sqrt{2} }{ \sqrt{x + 6} + \sqrt{2} } \right)=\frac{ x + 4 }{ \left(x + 4\right) \left(\sqrt{x + 6} + \sqrt{2}\right) } = \frac{1}{\sqrt{x + 6} + \sqrt{2}}  " /> The reduced function is continuous at  <img class="equation_image" title=" \displaystyle x = -4 " src="/equation_images/%20%5Cdisplaystyle%20x%20%3D%20-4%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{1}{\sqrt{x + 6} + \sqrt{2}} = \frac{\sqrt{2}}{4} " src="/equation_images/%20%5Cdisplaystyle%20%5Clim_%7Bx%20%5Cto%20-4%20%7D%5Cfrac%7B1%7D%7B%5Csqrt%7Bx%20%2B%206%7D%20%2B%20%5Csqrt%7B2%7D%7D%20%3D%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%20" alt="LaTeX:  \displaystyle \lim_{x \to -4 }\frac{1}{\sqrt{x + 6} + \sqrt{2}} = \frac{\sqrt{2}}{4} " data-equation-content=" \displaystyle \lim_{x \to -4 }\frac{1}{\sqrt{x + 6} + \sqrt{2}} = \frac{\sqrt{2}}{4} " /> </p> </p>