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

Question
Limits and continuity
Evaluate the following limits. $$\lim_{(x,y)\rightarrow(2,2)}\frac{y^2-4}{xy-2x}$$

2021-03-10
Evaluate:
$$\lim_{(x,y)\rightarrow(0,0)}\frac{y^2-4}{xy-2x}$$
Simplification:
$$\lim_{(x,y)\rightarrow(0,0)}\frac{y^2-4}{xy-2x}=\lim_{(x,y)\rightarrow(0,0)}\frac{y^2-2^2}{x(y-2)}$$
$$=\lim_{(x,y)\rightarrow(0,0)}\frac{(y+2)(y-2)}{x(y-2)}$$
$$=\lim_{(x,y)\rightarrow(0,0)}\frac{(y+2)}{x}$$
Apply the limit,
$$\lim_{(x,y)\rightarrow(0,0)}\frac{y^2-4}{xy-2x}=\frac{2+2}{2}$$
$$\frac{4}{2}$$
$$=2$$
Hence,
$$\lim_{(x,y)\rightarrow(0,0)}\frac{y^2-4}{xy-2x}=2$$

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