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

Evaluate the following limits.
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{yz-xy-xz-{x}^{2}}{yz+xy+xz-{y}^{2}}$
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Jozlyn
Evaluate:
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{yz-xy-xz-{x}^{2}}{yz+xy+xz-{y}^{2}}$
Simplification:
We have
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{yz-xy-xz-{x}^{2}}{yz+xy+xz-{y}^{2}}$
Apply the limit,
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{yz-xy-xz-{x}^{2}}{yz+xy+xz-{y}^{2}}=\frac{\left(1\right)\left(1\right)-\left(1\right)\left(1\right)-\left(1\right)\left(1\right)-\left(1{\right)}^{2}}{\left(1\right)\left(1\right)+\left(1\right)\left(1\right)+\left(1\right)\left(1\right)-\left(1{\right)}^{2}}$
$\frac{1-1-1-1}{1+1+1-1}$
$=\frac{-2}{2}$
$=-1$
Hence,
$\underset{\left(x,y,z\right)\to \left(1,1,1\right)}{lim}\frac{yz-xy-xz-{x}^{2}}{yz+xy+xz-{y}^{2}}=-1$
Jeffrey Jordon