"Evaluate the following limits. lim(x,y,z)→(1,1,1)x2+xy−xz−yzx−z" was answered by our experts

Zoe Oneal

Answered question

2021-02-05

Evaluate the following limits.

lim_{(x, y, z) \to (1, 1, 1)} \frac{x^{2} + x y - x z - y z}{x - z}

wornoutwomanC

Skilled2021-02-06Added 81 answers

We have to evaluate the limit of the function with three variable:

lim_{(x, y, z) \to (1, 1, 1)} \frac{x^{2} + x y - x z - y z}{x - z}

After putting value of limit we get that it is

\frac{0}{0}

form
We can solve this limit by laws of factorization.
Solving by factorization,

lim_{(x, y, z) \to (1, 1, 1)} \frac{x^{2} + x y - x z - y z}{x - z} = lim_{(x, y, z) \to (1, 1, 1)} \frac{x^{2} - x z + x y - y z}{x - z}

= lim_{(x, y, z) \to (1, 1, 1)} \frac{x (x - z) + y (x - z)}{x - z}

= lim_{(x, y, z) \to (1, 1, 1)} \frac{(x - z) (x + y)}{(x - z)}

= lim_{(x, y, z) \to (1, 1, 1)} (x + y)

= 1 + 1

= 2

Hence, value of given limit is 2.

Jeffrey Jordon

Expert2022-08-24Added 2605 answers

Answer is given below (on video)

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