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

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

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:
$$(\sqrt3+\sqrt3)$$
$$=2\sqrt3$$

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