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

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

Question
Limits and continuity
asked 2021-03-09
Evaluate the following limits. \(\lim_{(x,y)\rightarrow(2,2)}\frac{y^2-4}{xy-2x}\)

Answers (1)

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,
\(\lim_{(x,y)\rightarrow(0,0)}\frac{y^2-4}{xy-2x}=2\)
0

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