Question

Evaluate the following limits. \lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}+xy-xz-yz}{x-z}

Polynomial factorization
Evaluate the following limits.
$$\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}+xy-xz-yz}{x-z}$$

2021-05-13
Step 1
We have to evaluate the limit of the function with three variable:
$$\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}+xy-xz-yz}{x-z}$$
After putting value of limit we get that it is 00 form
We can solve this limit by laws of factorization.
Step 2
Solving by factorization,
$$\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}+xy-xz-yz}{x-z}=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}+xy-xz-yz}{x-z}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^{2}-xz+xy-yz}{x-z}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x(x-z)+y(x-z)}{x-z}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{(x-z)(x+y)}{(x-z)}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}(x+y)$$
=1+1
=2
Hence, value of given limit is 2.