Problem 56 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x-\sqrt{x z}-\sqrt{x y}+\sqrt{y z}}{x-\sqrt{x z}+\sqrt{x y}-\sqrt{y z}}$$

Step-by-Step Solution

Answer

If so, what is the value of the limit? $$\lim_{(x, y, z) \rightarrow(1,1,1)} \frac{x-\sqrt{xz}-\sqrt{xy}+\sqrt{yz}}{x-\sqrt{xz}+\sqrt{xy}-\sqrt{yz}}$$ Solution: Yes, the limit exists and is equal to -1.

Step 1: Analyze the given limit expression

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.

Step 2: Simplify the limit expression using algebraic manipulations

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: $$\lim_{(x, y, z) \rightarrow(1,1,1)} \frac{x-\sqrt{x z}-\sqrt{x y}+\sqrt{y z}}{x-\sqrt{x z}+\sqrt{x y}-\sqrt{y z}} \equiv \lim_{(x, y, z) \rightarrow(1,1,1)} \frac{(\sqrt{x y}-\sqrt{z})+(\sqrt{x z}-\sqrt{y})(\sqrt{y}+\sqrt{z})}{(\sqrt{x z}-\sqrt{y})+(\sqrt{x y}-\sqrt{z})(\sqrt{z}+\sqrt{y})}$$

Step 3: Cancel out common terms in the numerator and denominator

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.

Step 4: 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 \(-1\).