# Find the following limit: lim_{(x,y)rightarrow(3,3)}(frac{x-y}{sqrt x-sqrt y})

Find the following limit:
$\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}\right)$
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faldduE
Given:
$\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}\right)$
On putting the limits directly, we get:
$\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}\right)=\frac{0}{0}$
On further solving the equation, we have:
$\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\frac{x-y}{\sqrt{x}-\sqrt{y}}\right)=\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\frac{\left(\sqrt{x}{\right)}^{2}-\left(\sqrt{y}{\right)}^{2}}{\sqrt{x}-\sqrt{y}}\right)$
Since,
${a}^{2}-{b}^{2}=\left(a-b\right)\left(a+b\right)$
Therefore,
$\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}\right)$
$\underset{\left(x,y\right)\to \left(3,3\right)}{lim}\left(\sqrt{x}+\sqrt{y}\right)$
Now putting the limits, we get:
$\left(\sqrt{3}+\sqrt{3}\right)$
$=2\sqrt{3}$
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Jeffrey Jordon

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