Evaluate the following limits. lim_{(x,y)rightarrow(4,5)}frac{sqrt{x+y}-3}{x+y-9}

Evaluate the following limits. lim_{(x,y)rightarrow(4,5)}frac{sqrt{x+y}-3}{x+y-9}

Question
Limits and continuity
asked 2021-03-11
Evaluate the following limits.
\(\lim_{(x,y)\rightarrow(4,5)}\frac{\sqrt{x+y}-3}{x+y-9}\)

Answers (1)

2021-03-12
It is given that \(\lim_{(x,y)\rightarrow(4,5)}\frac{\sqrt{x+y}-3}{x+y-9}\)
\(\lim_{(x,y)\rightarrow(4,5)}\frac{\sqrt{x+y}-3}{x+y-9}=\lim_{(x,y)\rightarrow(4,5)}\frac{\sqrt{x+y}-3}{x+y-9}\times\frac{\sqrt{x+y}+3}{\sqrt{x+y}+3}\)
\(=\lim_{(x,y)\rightarrow(4,5)}\frac{(\sqrt{x+y})^2-3^2}{(x+y-9)(\sqrt{x+y}+3)}\)
\(=\lim_{(x,y)\rightarrow(4,5)}\frac{x+y-9}{(x+y-9)(\sqrt{x+y}+3)}\)
\(=\lim_{(x,y)\rightarrow(4,5)}\frac{1}{(\sqrt{x+y}+3)}\)
\(=\frac{1}{(\sqrt{4+5}+3)}\)
\(=\frac{1}{6}\)
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