Question

Evaluate
$\rm \displaystyle \lim_{x\to 0}\ \frac{\sqrt{x+2}-\sqrt{2}}{x}$

Answer 1

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Answer 2

Solution :

$\rm \displaystyle \lim_{x\to 0}\ \frac{\sqrt{x+2}-\sqrt{2}}{x}$

= $\rm \displaystyle \lim_{x\to 0}\ \frac{\sqrt{x+2}-\sqrt{2} \times (\sqrt{x+2}+\sqrt{2})}{x \times (\sqrt{x+2}+\sqrt{2})}$

=$\rm \displaystyle \lim_{x\to 0}\ \frac{(\sqrt{x+2})^{2}-(\sqrt{2})^{2}}{x \times (\sqrt{x+2}+\sqrt{2})}$

= $\rm \displaystyle \lim_{x\to 0}\ \frac{( x + 2 - 2 )}{x \times (\sqrt{x+2} + \sqrt{2})}$
= $\rm \displaystyle \frac{1}{(\sqrt{2+2}+ \sqrt{2})}$

= 1/(2 + √2 )

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