Find the following limit: lim_{(x,y)rightarrow(3,3)}(frac{x-y}{sqrt x-sqrt y})

Find the following limit: lim_{(x,y)rightarrow(3,3)}(frac{x-y}{sqrt x-sqrt y})

Question
Limits and continuity
asked 2021-02-21
Find the following limit:
\(\lim_{(x,y)\rightarrow(3,3)}(\frac{x-y}{\sqrt x-\sqrt y})\)

Answers (1)

2021-02-22
Given:
\(\lim_{(x,y)\rightarrow(3,3)}(\frac{x-y}{\sqrt x-\sqrt y})\)
On putting the limits directly, we get:
\(\lim_{(x,y)\rightarrow(3,3)}(\frac{x-y}{\sqrt x-\sqrt y})=\frac{0}{0}\)
On further solving the equation, we have:
\(\lim_{(x,y)\rightarrow(3,3)}(\frac{x-y}{\sqrt x-\sqrt y})=\lim_{(x,y)\rightarrow(3,3)}(\frac{(\sqrt x)^2-(\sqrt y)^2}{\sqrt x-\sqrt y})\)
Since,
\(a^2-b^2=(a-b)(a+b)\)
Therefore,
\(\lim_{(x,y)\rightarrow(3,3)}(\frac{(\sqrt x-\sqrt y)(\sqrt x+\sqrt y)}{\sqrt x-\sqrt y})\)
\(\lim_{(x,y)\rightarrow(3,3)}(\sqrt x+\sqrt y)\)
Now putting the limits, we get:
\((\sqrt3+\sqrt3)\)
\(=2\sqrt3\)
The required answer.
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