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

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