# Evaluate the following limits. lim_{(x,y)rightarrow(2,2)}frac{y^2-4}{xy-2x}

Evaluate the following limits. $\underset{\left(x,y\right)\to \left(2,2\right)}{lim}\frac{{y}^{2}-4}{xy-2x}$
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Gennenzip
Evaluate:
$\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{y}^{2}-4}{xy-2x}$
Simplification:
$\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{y}^{2}-4}{xy-2x}=\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{y}^{2}-{2}^{2}}{x\left(y-2\right)}$
$=\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{\left(y+2\right)\left(y-2\right)}{x\left(y-2\right)}$
$=\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{\left(y+2\right)}{x}$
Apply the limit,
$\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{y}^{2}-4}{xy-2x}=\frac{2+2}{2}$
$\frac{4}{2}$
$=2$
Hence,
$\underset{\left(x,y\right)\to \left(0,0\right)}{lim}\frac{{y}^{2}-4}{xy-2x}=2$
Jeffrey Jordon