Evaluate the following limit: _ x 2 x-2 x -2 - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}-2}{\sqrt{\text{x}}-\sqrt{2}}$

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Answer

$\lim\limits_{\text{x}\rightarrow2}\frac{\text{x}-2}{\sqrt{\text{x}}-\sqrt{2}}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}-2)\big(\sqrt{\text{x}}+\sqrt{2}\big)}{\big(\sqrt{\text{x}}-\sqrt{2}\big)\big(\sqrt{\text{x}}+\sqrt{2}\big)}$
$=\lim\limits_{\text{x}\rightarrow2}\frac{(\text{x}-2)\big(\sqrt{\text{x}}+\sqrt{2}\big)}{(\text{x}-2)}$
$=\sqrt{2}+\sqrt{2}$
$=2\sqrt{2}$

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