We are asked to evaluate the limit:

$\lim_{\left( x, y, z \right) \to \left( 4, 1, 1 \right)} \frac{z - \sqrt{zy} - \sqrt{zx} + \sqrt{xy}}{z - \sqrt{zy} + \sqrt{zx} - \sqrt{xy}}$

Step 1: Substitution of the limit point $(x, y, z) \to (4, 1, 1)$

We first attempt a direct substitution of $x = 4$, $y = 1$, and $z = 1$ into the given expression.

• $z = 1$
• $\sqrt{zy} = \sqrt{1 \cdot 1} = 1$
• $\sqrt{zx} = \sqrt{1 \cdot 4} = 2$
• $\sqrt{xy} = \sqrt{4 \cdot 1} = 2$

Substituting these values into the numerator and the denominator:

Numerator:

$z - \sqrt{zy} - \sqrt{zx} + \sqrt{xy} = 1 - 1 - 2 + 2 = 0$

Denominator:

$z - \sqrt{zy} + \sqrt{zx} - \sqrt{xy} = 1 - 1 + 2 - 2 = 0$

Both the numerator and denominator result in 0, so we have a $0/0$ indeterminate form. Thus, we can apply L'Hopital's Rule or attempt to simplify the expression further.

Step 2: Simplification

We attempt to simplify the expression by expanding and combining terms. The expression is symmetric in form, with similar terms in both the numerator and denominator. Let's analyze the structure carefully to see if common terms can be factored or canceled out.

Let me simplify the expression further to evaluate the limit.It seems there was a mistake in my approach to evaluating the limit with multiple variables at once. I'll fix this by computing the limit step by step, evaluating first as $z \to 1$, then $y \to 1$, and finally $x \to 4$.

Let me handle the computations properly.The limit of the given expression as $(x, y, z) \to (4, 1, 1)$ is:

$\lim_{\left( x, y, z \right) \to \left( 4, 1, 1 \right)} \frac{z - \sqrt{zy} - \sqrt{zx} + \sqrt{xy}}{z - \sqrt{zy} + \sqrt{zx} - \sqrt{xy}} = -\frac{1}{3}$

So, the correct answer is:

A. $-\frac{1}{3}$.

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Would you like more details on the steps, or do you have any questions?

Here are five related questions you might find helpful:

1. How do we apply L'Hopital's Rule in evaluating indeterminate forms like $0/0$?
2. What are other techniques to simplify complex limits involving square roots?
3. How can partial limits (like fixing one variable) help in multi-variable limit evaluation?
4. What does symmetry in expressions tell us about limits in calculus?
5. What are common pitfalls when substituting values in limit problems?

Tip: In multi-variable calculus, simplifying terms before applying limits can often reveal underlying structures or symmetries that make the limit easier to calculate.