lim x → 2 √ x − 2 + √ x − √ 2 √ x 2 − 4 is equal to Options: A . &nbsp 1 2 B . &nbsp 1 C . &nbsp 2 D . &nbsp 0

Answer: Option A : A lim x → 2 √ x − 2 + √ x − √ 2 √ x 2 − 4 = lim x → 2 ( √ x − 2 √ x + 2 √ x − 2 + √ x − √ 2 √ x 2 − 4 ) On rationalisation - = lim x → 2 ( 1 √ x + 2 + x − 2 √ x 2 − 4 ( √ x + √ 2 ) ) = lim x → 2 1 √ x + 2 + lim x → 2 √ x − 2 x + 2 × 1 √ x + √ 2 = 1 2

limx→2√x−2+√x−√2√x2−4

Discussion Forum : Limits Continuity And Differentiability

Question -

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

Options:

A .

\frac{1}{2}

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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