Question: "Find the following limit: lim(x,y)→(3,3)(x−yx−y)"

sibuzwaW

Answered question

2021-02-21

Find the following limit:

lim_{(x, y) \to (3, 3)} (\frac{x - y}{\sqrt{x} - \sqrt{y}})

faldduE

Skilled2021-02-22Added 109 answers

Given:

lim_{(x, y) \to (3, 3)} (\frac{x - y}{\sqrt{x} - \sqrt{y}})

On putting the limits directly, we get:

lim_{(x, y) \to (3, 3)} (\frac{x - y}{\sqrt{x} - \sqrt{y}}) = \frac{0}{0}

On further solving the equation, we have:

lim_{(x, y) \to (3, 3)} (\frac{x - y}{\sqrt{x} - \sqrt{y}}) = lim_{(x, y) \to (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) \to (3, 3)} (\frac{(\sqrt{x} - \sqrt{y}) (\sqrt{x} + \sqrt{y})}{\sqrt{x} - \sqrt{y}})

lim_{(x, y) \to (3, 3)} (\sqrt{x} + \sqrt{y})

Now putting the limits, we get:

(\sqrt{3} + \sqrt{3})

= 2 \sqrt{3}

The required answer.

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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