91Ó°ÊÓ

Notice that, as the limit approaches (1,1,1), the denominator approaches 0, so the limit will be an indeterminate form if the numerator approaches 0 as well. To deal with indeterminate forms, we need to find a way to simplify the expression.

We will factor the numerator and denominator to cancel out common terms. To do this, we will use the technique called difference of squares: Notice that \((a+b)(a-b)=a^2-b^2\). Now we can rewrite the limit as: $$\lim_{(x, y, z) \rightarrow(1,1,1)} \frac{[(\sqrt{x y}+\sqrt{z})+(\sqrt{x z}-\sqrt{y})(\sqrt{y}-\sqrt{z})]}{[(\sqrt{x z}+\sqrt{y})+(\sqrt{x y}-\sqrt{z})(\sqrt{z}-\sqrt{y})]}$$ Now use \((a-b)(a+b)=a^2-b^2\) to simplify the terms: $$\lim_{(x, y, z) \rightarrow(1,1,1)} \frac{x-\sqrt{x z}-\sqrt{x y}+\sqrt{y z}}{x-\sqrt{x z}+\sqrt{x y}-\sqrt{y z}} \equiv \lim_{(x, y, z) \rightarrow(1,1,1)} \frac{(\sqrt{x y}-\sqrt{z})+(\sqrt{x z}-\sqrt{y})(\sqrt{y}+\sqrt{z})}{(\sqrt{x z}-\sqrt{y})+(\sqrt{x y}-\sqrt{z})(\sqrt{z}+\sqrt{y})}$$

Observe that we can rewrite the denominator as: \((\sqrt{x y}-\sqrt{z})(-\sqrt{z}-\sqrt{y})+(\sqrt{x z}-\sqrt{y})(\sqrt{y}+\sqrt{z}) = -(\sqrt{x y}-\sqrt{z})(\sqrt{z}+\sqrt{y})\) So the limit simplifies to: $$\lim_{(x, y, z) \rightarrow(1,1,1)} \frac{(\sqrt{x y}-\sqrt{z})+(\sqrt{x z}-\sqrt{y})(\sqrt{y}+\sqrt{z})}{-(\sqrt{x y}-\sqrt{z})(\sqrt{z}+\sqrt{y})+(\sqrt{x z}-\sqrt{y})(\sqrt{y}+\sqrt{z})} = -1$$ Having simplified the expression, we can evaluate the limit.

Now that the expression is simplified, we can find the limit as \((x, y, z) \rightarrow(1,1,1)\): $$\lim_{(x, y, z) \rightarrow(1,1,1)} (-1) = -1$$ So, the limit exists and is equal to \(-1\).