Problem 55 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}}$$

Step-by-Step Solution

Answer

Answer: The value of the limit is $-\frac{1}{2}$.

Step 1: Write down the given limit

We are given the limit: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}}$$

Step 2: Factor the numerator and denominator

We will now try to factor the terms in the numerator and denominator. First, observe that the terms in the numerator have either x or y as a common factor. Similarly, the terms in the denominator have either x or y as a common factor. Factoring out x and y from the numerator and denominator respectively, we get: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{x(yz - y - z - x)}{y(xz + x + z - y)}$$

Step 3: Cancel out the common factors

Since the numerator and the denominator share common factors x and y, we can cancel out those factors to simplify the fraction: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{(yz - y - z - x)}{(xz + x + z - y)}$$

Step 4: Find the limit

Now that we have simplified the fraction, we can substitute the limits for x, y, and z to find the limit: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{(yz - y - z - x)}{(xz + x + z - y)} = \frac{(1 \cdot 1 - 1 - 1 - 1)}{(1 \cdot 1 + 1 + 1 - 1)} = \frac{-1}{2}$$ The limit of the given function is therefore: $$\lim _{(x, y, z) \rightarrow(1,1,1)} \frac{y z-x y-x z-x^{2}}{y z+x y+x z-y^{2}} = -\frac{1}{2}$$