Evaluate the following limits.

$\underset{(x,y,z)\to (1,1,1)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}$

palmantkf4u
2022-03-30
Answered

Evaluate the following limits.

$\underset{(x,y,z)\to (1,1,1)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}$

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sorrisi7yny

Answered 2022-03-31
Author has **9** answers

Given:

$\underset{(x,y,z)\to (1,1,1)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}$

On simplification, we get:

$\underset{(x,y,z)\to (1,1,1)}{lim}\frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}=\underset{(x,y,z)\to (1,1,1)}{lim}\frac{\sqrt{x}(\sqrt{x}-\sqrt{z})-\sqrt{y}(\sqrt{x}-\sqrt{z})}{\sqrt{x}(\sqrt{x}-\sqrt{z})+\sqrt{y}(\sqrt{x}-\sqrt{z})}$

$=\underset{(x,y,z)\to (1,1,1)}{lim}\frac{(\sqrt{x}-\sqrt{z})(\sqrt{x}-\sqrt{y})}{(\sqrt{x}-\sqrt{z})(\sqrt{x}+\sqrt{y})}$

$=\underset{(x,y,z)\to (1,1,1)}{lim}\frac{(\sqrt{x}-\sqrt{y})}{(\sqrt{x}+\sqrt{y})}$

$\frac{(\sqrt{1}-\sqrt{1})}{(\sqrt{1}+\sqrt{1})}$

$=0$

On simplification, we get:

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