91Ó°ÊÓ

As \((x, y, z) \rightarrow (1, 1, 1)\), we notice that both the numerator and denominator of the expression approach 0: $$\frac{1-\sqrt{1}-\sqrt{1}+\sqrt{1}}{1-\sqrt{1}+\sqrt{1}-\sqrt{1}} = \frac{0}{0}$$ Since both the numerator and denominator approach 0, further algebraic manipulation is needed in order to evaluate the limit.

To evaluate the limit, we will try to simplify the expression by multiplying the numerator and denominator by the conjugate of the given expression. The conjugate of the given expression is: $$(x+\sqrt{x z}-\sqrt{x y}+\sqrt{y z})$$ By multiplying the numerator and denominator with the conjugate, we get: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{(x-\sqrt{x z}-\sqrt{x y}+\sqrt{y z})(x+\sqrt{x z}-\sqrt{x y}+\sqrt{y z})}{(x-\sqrt{x z}+\sqrt{x y}-\sqrt{y z})(x+\sqrt{x z}-\sqrt{x y}+\sqrt{y z})}$$ Now we must analyze what this multiplication will result in for our expression: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x^2 - xz - xy + yz}{x^2 - xz + xy - yz}$$ As multiplication does not change the limit, we can further simplify this expression by canceling the terms \(xz\) and \(-yz\) from both the numerator and denominator: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x^2 - xy}{x^2 + xy}$$

Now let's plug in the values \((1,1,1)\) into our simplified expression: $$\lim_{(x, y, z) \rightarrow (1, 1, 1)} \frac{x^2 - xy}{x^2 + xy} = \frac{1 - 1}{1+1} = \frac{0}{2} = 0$$

The limit of the given expression as \((x, y, z) \rightarrow (1,1,1)\) is 0.