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

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

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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