palmantkf4u

2022-03-30

Evaluate the following limits.
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}$

sorrisi7yny

Given:
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}$
On simplification, we get:
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}=\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{\sqrt{x}\left(\sqrt{x}-\sqrt{z}\right)-\sqrt{y}\left(\sqrt{x}-\sqrt{z}\right)}{\sqrt{x}\left(\sqrt{x}-\sqrt{z}\right)+\sqrt{y}\left(\sqrt{x}-\sqrt{z}\right)}$
$=\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}-\sqrt{z}\right)\left(\sqrt{x}+\sqrt{y}\right)}$
$=\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{\left(\sqrt{x}-\sqrt{y}\right)}{\left(\sqrt{x}+\sqrt{y}\right)}$
$\frac{\left(\sqrt{1}-\sqrt{1}\right)}{\left(\sqrt{1}+\sqrt{1}\right)}$
$=0$

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