Question

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

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

2021-02-06
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 $$\frac{0}{0}$$ form
We can solve this limit by laws of factorization.
Solving by factorization,
$$\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^2+xy-xz-yz}{x-z}=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x^2-xz+xy-yz}{x-z}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{x(x-z)+y(x-z)}{x-z}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}\frac{(x-z)(x+y)}{(x-z)}$$
$$=\lim_{(x,y,z)\rightarrow(1,1,1)}(x+y)$$
$$=1+1$$
$$=2$$
Hence, value of given limit is 2.