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

Question

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

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Answer

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

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