Evaluate the following limit: _ x 0 1+x - 1-x 2x - Vidyadip

Question

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{1+\text{x}}-\sqrt{1-\text{x}}}{2\text{x}}$

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Answer

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

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