# Evaluate the following limits. lim_{(x,y)rightarrow(4,5)}frac{sqrt{x+y}-3}{x+y-9}

Evaluate the following limits.
$\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{\sqrt{x+y}-3}{x+y-9}$
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Viktor Wiley
It is given that $\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{\sqrt{x+y}-3}{x+y-9}$
$\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{\sqrt{x+y}-3}{x+y-9}=\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{\sqrt{x+y}-3}{x+y-9}×\frac{\sqrt{x+y}+3}{\sqrt{x+y}+3}$
$=\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{\left(\sqrt{x+y}{\right)}^{2}-{3}^{2}}{\left(x+y-9\right)\left(\sqrt{x+y}+3\right)}$
$=\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{x+y-9}{\left(x+y-9\right)\left(\sqrt{x+y}+3\right)}$
$=\underset{\left(x,y\right)\to \left(4,5\right)}{lim}\frac{1}{\left(\sqrt{x+y}+3\right)}$
$=\frac{1}{\left(\sqrt{4+5}+3\right)}$
$=\frac{1}{6}$
Jeffrey Jordon