Evaluate the following limits. lim_{(x,y,z)rightarrow(1,-1,1)}frac{xz+5x+yz+5y}{x+y}

We have to find limits of the functions:
$\underset{(x,y,z)\to (1,-1,1)}{lim}\frac{xz+5x+yz+5y}{x+y}$
As it is $\frac{0}{0}$ indeterminate form so solving by factorizing of numerator as well as denominator, we get
$\underset{(x,y,z)\to (1,-1,1)}{lim}\frac{xz+5x+yz+5y}{x+y}=\underset{(x,y,z)\to (1,-1,1)}{lim}\frac{x(z+5)+y(z+5)}{x+y}$
$=\underset{(x,y,z)\to (1,-1,1)}{lim}\frac{(x+y)(z+5)}{(x+y)}$
$=\underset{(x,y,z)\to (1,-1,1)}{lim}(z+5)$
Putting x=1, y=-1, x=1, we get
$=\underset{(x,y,z)\to (1,-1,1)}{lim}(z+5)=(1+5)$
$=6$
Hence, value of limit is 6.