Evaluate the following limit: _ x 0 2-x - 2+x x - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{2-\text{x}}-\sqrt{2+\text{x}}}{\text{x}}$

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Answer

$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{2-\text{x}}-\sqrt{2+\text{x}}}{\text{x}}$$=\lim\limits_{\text{x}\rightarrow0}\frac{\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}{\text{x}\times\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{(2-\text{x})-(2+\text{x})}{\text{x}\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}$
$=\lim\limits_{\text{x}\rightarrow0}\frac{-2\text{x}}{\text{x}\big(\sqrt{2-\text{x}}+\sqrt{2+\text{x}}\big)}$
$=\frac{-2}{\sqrt{2}+\sqrt{2}}$
$=\frac{-2}{2\sqrt{2}}$
$=\frac{-1}{\sqrt{2}}$

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