limx→2√x−2+√x−√2√x2−4 is

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Question

lim x \to 2 \frac{\sqrt{x - 2} + \sqrt{x} - \sqrt{2}}{\sqrt{x^{2} - 4}}

is equal to

Options:

A . 12

B . 1

C . 2

D . 0

Answer: Option A
:
A

lim x \to 2 \frac{\sqrt{x - 2} + \sqrt{x} - \sqrt{2}}{\sqrt{x^{2} - 4}}

= lim x \to 2 (\frac{\sqrt{x - 2}}{\sqrt{x + 2} \sqrt{x - 2}} + \frac{\sqrt{x} - \sqrt{2}}{\sqrt{x^{2} - 4}})

On rationalisation -

= lim x \to 2 (\frac{1}{\sqrt{x + 2}} + \frac{x - 2}{\sqrt{x^{2} - 4} (\sqrt{x} + \sqrt{2})})

= lim x \to 2 \frac{1}{\sqrt{x + 2}} + lim x \to 2 \sqrt{\frac{x - 2}{x + 2}} \times \frac{1}{\sqrt{x} + \sqrt{2}}

= \frac{1}{2}

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